diff --git a/previews/PR184/.documenter-siteinfo.json b/previews/PR184/.documenter-siteinfo.json new file mode 100644 index 00000000..132b23a0 --- /dev/null +++ b/previews/PR184/.documenter-siteinfo.json @@ -0,0 +1 @@ +{"documenter":{"julia_version":"1.11.1","generation_timestamp":"2024-11-26T02:03:17","documenter_version":"1.8.0"}} \ No newline at end of file diff --git a/previews/PR184/API/index.html b/previews/PR184/API/index.html new file mode 100644 index 00000000..19ce2530 --- /dev/null +++ b/previews/PR184/API/index.html @@ -0,0 +1,40 @@ + +API · NetworkDynamics
GitHub

API

The following functions are designed for public use.

Network Construction API

NetworkDynamics.Network — Type
Network([g,] vertexf, edgef; kwarg...)

Construct a Network object from a graph g and edge and component models vertexf and edgef.

Arguments:

  • g::AbstractGraph: The graph on which the network is defined. Optional, can be ommittet if all component models have a defined graphelement. See vidx and src/dst keywors for VertexModel and EdgeModel constructors respectively.

  • vertexm: A single VertexModel or a vector of VertexModel objects. The order of the vertex models must mirror the order of the vertices(g) iterator.

  • edgem: A single EdgeModel or a vector of EdgeModel objects. The order of the edge models must mirror the order of the edges(g) iterator.

Optional keyword arguments:

  • execution=SequentialExecution{true}(): Execution model of the network. E.g. SequentialExecution, KAExecution, PolyesterExecution or ThreadedExecution.
  • aggregator=execution isa SequentialExecution ? SequentialAggregator(+) : PolyesterAggregator(+): Aggregation function applied to the edge models. E.g. SequentialAggregator, PolyesterAggregator, ThreadedAggregator, SparseAggregator.
  • check_graphelement=true: Check if the graphelement metadata is consistent with the graph.
  • dealias=false Check if the components alias eachother and create copies if necessary. This is necessary if the same component model is referenced in multiple places in the Network but you want to dynamicially asign metadata, such as initialization information to specific instances.
  • verbose=false: Show additional information during construction.
source
Network(nw::Network; g, vertexm, edgem, kwargs...)

Rebuild the Network with same graph and vertex/edge models but possibly different kwargs.

source
NetworkDynamics.dim — Method
dim(nw::Network)

Returns the number of dynamic states in the network, corresponts to the length of the flat state vector.

source
NetworkDynamics.pdim — Method
pdim(nw::Network)

Returns the number of parameters in the network, corresponts to the length of the flat parameter vector.

source

Component Models

NetworkDynamics.VertexModel — Method
VertexModel(; kwargs...)

Build a VertexModel according to the keyword arguments.

Main Arguments:

  • f=nothing: Dynamic function of the component. Can be nothing if dim is 0.
  • g: Output function of the component. Usefull helpers: StateMask
  • sym/dim: Symbolic names of the states. If dim is provided, sym is set automaticially.
  • outsym/outdim: Symbolic names of the outputs. If outdim is provided, outsym is set automaticially. Can be infered automaticially if g isa StateMask.
  • psym/pdim=0: Symbolic names of the parameters. Ifpdimis provided,psym` is set automaticially.
  • mass_matrix=I: Mass matrix of component. Can be a vector v and is then interpreted as Diagonal(v).
  • name=dim>0 ? :VertexM : :StaticVertexM: Name of the component.

Optional Arguments:

  • insym/indim: Symbolic names of the inputs. If indim is provided, insym is set automaticially.
  • vidx: Index of the vertex in the graph, enables graphless constructor.
  • ff: FeedForwardType of component. Will be typically infered from g automaticially.
  • obssym/obsf: Define additional "observable" states.
  • symmetadata/metadata: Provide prefilled metadata dictionaries.

All Symbol arguments can be used to set default values, i.e. psym=[:K=>1, :p].

source
NetworkDynamics.EdgeModel — Method
EdgeModel(; kwargs...)

Build a EdgeModel according to the keyword arguments.

Main Arguments:

  • f=nothing: Dynamic function of the component. Can be nothing if dim is 0.
  • g: Output function of the component. Usefull helpers: AntiSymmetric, Symmetric, Fiducial, Directed and StateMask.
  • sym/dim: Symbolic names of the states. If dim is provided, sym is set automaticially.
  • outsym/outdim: Symbolic names of the outputs. If outdim is provided, outsym is set automaticially. In general, outsym for edges isa named tuple (; src, dst). However, depending on the g function, it might be enough to provide a single vector or even nothing (e.g. AntiSymmetric(StateMask(1:2))). See Building EdgeModels for examples.
  • psym/pdim=0: Symbolic names of the parameters. Ifpdimis provided,psym` is set automaticially.
  • mass_matrix=I: Mass matrix of component. Can be a vector v and is then interpreted as Diagonal(v).
  • name=dim>0 ? :EdgeM : :StaticEdgeM: Name of the component.

Optional Arguments:

  • insym/indim: Symbolic names of the inputs. If indim is provided, insym is set automaticially. For edges, insym is a named tuple (; src, dst). If give as vector tuple is created automaticially.
  • src/dst: Index or name of the vertices at src and dst end. Enables graphless constructor.
  • ff: FeedForwardType of component. Will be typically infered from g automaticially.
  • obssym/obsf: Define additional "observable" states.
  • symmetadata/metadata: Provide prefilled metadata dictionaries.

All Symbol arguments can be used to set default values, i.e. psym=[:K=>1, :p].

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Component Models with MTK

NetworkDynamics.VertexModel — Method
VertexModel(sys::ODESystem, inputs, outputs; ff_to_constraint=true, kwargs...)

Create a VertexModel object from a given ODESystem created with ModelingToolkit. You need to provide 2 lists of symbolic names (Symbol or Vector{Symbols}):

  • inputs: names of variables in you equation representing the aggregated edge states
  • outputs: names of variables in you equation representing the node output

ff_to_constraint controlls, whether output transformations g which depend on inputs should be

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NetworkDynamics.EdgeModel — Method
EdgeModel(sys::ODESystem, srcin, srcout, dstin, dstout; ff_to_constraint=false, kwargs...)

Create a EdgeModel object from a given ODESystem created with ModelingToolkit. You need to provide 4 lists of symbolic names (Symbol or Vector{Symbols}):

  • srcin: names of variables in you equation representing the node state at the source
  • dstin: names of variables in you equation representing the node state at the destination
  • srcout: names of variables in you equation representing the output at the source
  • dstout: names of variables in you equation representing the output at the destination

ff_to_constraint controlls, whether output transformations g which depend on inputs should be transformed into constraints.

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NetworkDynamics.EdgeModel — Method
EdgeModel(sys::ODESystem, srcin, dstin, AntiSymmetric(dstout); ff_to_constraint=false, kwargs...)

Create a EdgeModel object from a given ODESystem created with ModelingToolkit.

Here you only need to provide one list of output symbols: dstout. To make it clear how to handle the single-sided output definiton, you musst wrap the symbol vector in

  • AntiSymmetric(dstout),
  • Symmetric(dstout), or
  • Directed(dstout).

ff_to_constraint controlls, whether output transformations g which depend on inputs should be transformed into constraints.

source

Output Function Helpers/Wrappers

NetworkDynamics.StateMask — Type
StateMask(i::AbstractArray)
+StateMaks(i::Number)

A StateMask is a predefined output function. It can be used to define the output of a component model by picking from the internal state.

I.e. g=StateMask(2:3) in a vertex function will output the internal states 2 and 3. In many contexts, StateMasks can be constructed implicitly by just providing the indices, e.g. g=1:2.

For EdgeModel this needs to be combined with a Directed, Symmetric, AntiSymmetric or Fiducial coupling, e.g. g=Fiducial(1:2, 3:4) forwards states 1:2 to dst and states 3:4 to src.

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Accessors for Component Properties

NetworkDynamics.outdim — Function
outdim(c::VertexModel)::Int
+outdim(c::EdgeModel)::@NamedTuple(src::Int, dst::Int)

Retrieve the output dimension of the component

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NetworkDynamics.outsym — Function

outsym(c::VertexModel)::Vector{Symbol} outsym(c::EdgeModel)::@NamedTuple{src::Vector{Symbol}, dst::Vector{Symbol}}

Retrieve the output symbols of the component.

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NetworkDynamics.insym — Function
insym(c::VertexModel)::Vector{Symbol}
+insym(c::EdgeModel)::@NamedTuple{src::Vector{Symbol}, dst::Vector{Symbol}}

Musst be called after hasinsym/hasindim returned true. Gives the insym vector(s). For vertex model just a single vector, for edges it returns a named tuple (; src, dst) with two symbol vectors.

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FeedForwardType-Traits

Symbolic Indexing API

Network Parameter Object

NetworkDynamics.NWParameter — Type
NWParameter(nw_or_nw_wraper, pflat)

Indexable wrapper for flat parameter array pflat. Needs Network or wrapper of Network, e.g. ODEProblem.

p = NWParameter(nw)
+p.v[idx, :sym] # get parameter :sym of vertex idx
+p.e[idx, :sym] # get parameter :sym of edge idx
+p[s::Union{VPIndex, EPIndex}] # get parameter for specific index

Get flat array representation using pflat(p).

source
NetworkDynamics.NWParameter — Method
NWParameter(nw_or_nw_wraper;
+            ptype=Vector{Float64}, pfill=filltype(ptype), default=true)

Creates "empty" NWParameter object for the Network/Wrapper nw with flat type ptype. The array will be prefilled with pfill (defaults to NaN).

If default=true the default parameter values attached to the network components will be loaded.

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Network State Object

NetworkDynamics.NWState — Type
NWState(nw_or_nw_wrapper, uflat, [pflat], [t])

Indexable wrapper for flat state & parameter array. Needs Network or wrapper of Network, e.g. ODEProblem.

s = NWState(nw)
+s.v[idx, :sym] # get state :sym of vertex idx
+s.e[idx, :sym] # get state :sym of edge idx
+s.p.v[idx, :sym] # get parameter :sym of vertex idx
+s.p.e[idx, :sym] # get parameter :sym of edge idx
+s[s::Union{VIndex, EIndex, EPIndex, VPIndex}] # get parameter for specific index

Get flat array representation using uflat(s) and pflat(s).

source
NetworkDynamics.NWState — Method
NWState(nw_or_nw_wrapper;
+        utype=Vector{Float64}, ufill=filltype(utype),
+        ptype=Vector{Float64}, pfill=filltype(ptype), default=true)

Creates "empty" NWState object for the Network/Wrapper nw with flat types utype & ptype. The arrays will be prefilled with ufill and pfill respectively (defaults to NaN).

If default=true the default state & parameter values attached to the network components will be loaded.

source
NetworkDynamics.NWState — Method
NWState(p::NWState; utype=typeof(uflat(s)), ptype=typeof(pflat(s)))

Create NWState based on other state object, just convert types.

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NetworkDynamics.NWState — Method
NWState(p::NWParameter; utype=Vector{Float64}, ufill=filltype(utype), default=true)

Create NWState based on existing NWParameter object.

source
NetworkDynamics.pflat — Function
pflat(p::NWParameter)
+pflat(s::NWState)

Retrieve the wrapped flat array representation of the parameters.

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Symbolic Indices

NetworkDynamics.VIndex — Type
VIndex{C,S} <: SymbolicStateIndex{C,S}
+idx = VIndex(comp, sub)

A symbolic index for a vertex state variable.

  • comp: the component index, either int or a collection of ints
  • sub: the subindex, either int, symbol or a collection of those.
VIndex(1, :P)      # vertex 1, variable :P
+VIndex(1:5, 1)     # first state of vertices 1 to 5
+VIndex(7, (:x,:y)) # states :x and :y of vertex 7

Can be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWState, NWParameter or ODESolution.

See also: EIndex, VPIndex, EPIndex

source
NetworkDynamics.EIndex — Type
EIndex{C,S} <: SymbolicStateIndex{C,S}
+idx = EIndex(comp, sub)

A symbolic index for an edge state variable.

  • comp: the component index, either int or a collection of ints
  • sub: the subindex, either int, symbol or a collection of those.
EIndex(1, :P)      # edge 1, variable :P
+EIndex(1:5, 1)     # first state of edges 1 to 5
+EIndex(7, (:x,:y)) # states :x and :y of edge 7

Can be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWState, NWParameter or ODESolution.

See also: VIndex, VPIndex, EPIndex

source
NetworkDynamics.VPIndex — Type
VPIndex{C,S} <: SymbolicStateIndex{C,S}
+idx = VPIndex(comp, sub)

A symbolic index into the parameter a vertex:

  • comp: the component index, either int or a collection of ints
  • sub: the subindex, either int, symbol or a collection of those.

Can be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWParameter or ODEProblem.

See also: EPIndex, VIndex, EIndex

source
NetworkDynamics.EPIndex — Type
EPIndex{C,S} <: SymbolicStateIndex{C,S}
+idx = VEIndex(comp, sub)

A symbolic index into the parameter a vertex:

  • comp: the component index, either int or a collection of ints
  • sub: the subindex, either int, symbol or a collection of those.

Can be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWParameter or ODEProblem.

See also: VPIndex, VIndex, EIndex

source

Index generators

NetworkDynamics.vidxs — Function
vidxs([inpr], components=:, variables=:) :: Vector{VIndex}

Generate vector of symbolic indexes for vertices.

  • inpr: Only needed for name matching or : access. Can be Network, sol, prob, ...
  • components: Number/Vector, :, Symbol (name matches), String/Regex (name contains)
  • variables: Symbol/Number/Vector, :, String/Regex (all sym containing)

Examples:

vidxs(nw)                 # all vertex state indices
+vidxs(1:2, :u)            # [VIndex(1, :u), VIndex(2, :u)]
+vidxs(nw, :, [:u, :v])    # [VIndex(i, :u), VIndex(i, :v) for i in 1:nv(nw)]
+vidxs(nw, "ODEVertex", :) # all symbols of all vertices with name containing "ODEVertex"
source
NetworkDynamics.eidxs — Function
vidxs([inpr], components=:, variables=:) :: Vector{EIndex}

Generate vector of symbolic indexes for edges.

  • inpr: Only needed for name matching or : access. Can be Network, sol, prob, ...
  • components: Number/Vector, :, Symbol (name matches), String/Regex (name contains)
  • variables: Symbol/Number/Vector, :, String/Regex (all sym containing)

Examples:

eidxs(nw)                # all edge state indices
+eidxs(1:2, :u)           # [EIndex(1, :u), EIndex(2, :u)]
+eidxs(nw, :, [:u, :v])   # [EIndex(i, :u), EIndex(i, :v) for i in 1:ne(nw)]
+eidxs(nw, "FlowEdge", :) # all symbols of all edges with name containing "FlowEdge"
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NetworkDynamics.vpidxs — Function
vpidxs([inpr], components=:, variables=:) :: Vector{VPIndex}

Generate vector of symbolic indexes for parameters. See vidxs for more information.

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NetworkDynamics.epidxs — Function
epidxs([inpr], components=:, variables=:) :: Vector{EPIndex}

Generate vector of symbolic indexes for parameters. See eidxs for more information.

source

Metadata API

Component Metadata API

NetworkDynamics.get_graphelement — Function
get_graphelement(c::EdgeModel)::@NamedTuple{src::T, dst::T}
+get_graphelement(c::VertexModel)::Int

Retrieves the graphelement metadata for the component model. For edges this returns a named tupe (;src, dst) where both are either integers (vertex index) or symbols (vertex name).

source
NetworkDynamics.set_graphelement! — Function
set_graphelement!(c::EdgeModel, src, dst)
+set_graphelement!(c::VertexModel, vidx)

Sets the graphelement metadata for the edge model. For edges this takes two arguments src and dst which are either integer (vertex index) or symbol (vertex name). For vertices it takes a single integer vidx.

source

Per-Symbol Metadata API

NetworkDynamics.set_metadata! — Method
set_metadata!(c::ComponentModel, sym::Symbol, key::Symbol, value)
+set_metadata!(c::ComponentModel, sym::Symbol, pair)

Sets the metadata key for symbol sym to value.

source

Initialization

NetworkDynamics.find_fixpoint — Function
find_fixpoint(nw::Network, [x0::NWState=NWState(nw)], [p::NWParameter=x0.p]; kwargs...)
+find_fixpoint(nw::Network, x0::AbstractVector, p::AbstractVector; kwargs...)
+find_fixpoint(nw::Network, x0::AbstractVector; kwargs...)

Convenience wrapper around SteadyStateProblem from SciML-ecosystem. Constructs and solves the steady state problem, returns found value wrapped as NWState.

source
NetworkDynamics.initialize_component! — Function
initialize_component!(cf::ComponentModel; verbose=true, kwargs...)

Initialize a ComponentModel by solving the corresponding NonlinearLeastSquaresProblem. During initialization, everyting which has a default value (see Metadata) is considered "fixed". All other variables are considered "free" and are solved for. The initial guess for each variable depends on the guess value in the Metadata.

The result is stored in the ComponentModel itself. The values of the free variables are stored in the metadata field init.

The kwargs are passed to the nonlinear solver.

source
NetworkDynamics.init_residual — Function
init_residual(cf::T; t=NaN, recalc=false)

Calculates the residual |du| for the given component model for the values provided via default and init Metadata.

If recalc=false just return the residual determined in the actual initialization process.

See also initialize_component!.

source

Execution Types

NetworkDynamics.ExecutionStyle — Type
abstract type ExecutionStyle{buffered::Bool} end

Abstract type for execution style. The coreloop dispatches based on the Execution style stored in the network object.

  • buffered=true means that the edge input es explicitly gathered, i.e. the vertex outputs in the output buffer will be copied into a dedicated input buffer for the edges.
  • buffered=false means, that the edge inputs are not explicitly gathered, but the corloop will perform a redirected lookup into the output buffer.
source

Aggregators

NetworkDynamics.Aggregator — Type
abstract type Aggregator end

Abstract sypertype for aggregators. Aggregators operate on the output buffer of all components and fill the aggregation buffer with the aggregatated edge values per vertex.

All aggregators have the constructor

Aggegator(aggfun)

for example

SequentialAggreator(+)
source

Utils

NetworkDynamics.save_parameters! — Function
save_parameters!(integrator::SciMLBase.DEIntegrator)

Save the current parameter values in the integrator. Call this function inside callbacks if the parameter values have changed. This will store a timeseries of said parameters in the solution object, thus alowing us to recosntruct observables which depend on time-dependet variables.

source
NetworkDynamics.ff_to_constraint — Function
ff_to_constraint(v::VertexModel)

Takes VertexModel v with feed forward and turns all algebraic output states into internal states by defining algebraic constraints contraints 0 = out - g(...). The new output function is just a StateMask into the extended internal state vector.

Returns the transformed VertexModel.

source
Base.copy — Method
copy(c::NetworkDynamics.ComponentModel)

Shallow copy of the component model. Creates a deepcopy of metadata and symmetadata but references the same objects everywhere else.

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+$(document).ready(function () { + $("#documenter .docs-navbar").headroom({ + tolerance: { up: 10, down: 10 }, + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let meta = $("div[data-docstringscollapsed]").data(); + + if (meta?.docstringscollapsed) { + $("#documenter-article-toggle-button").trigger({ + type: "click", + noToggleAnimation: true, + }); + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +/* +To get an in-depth about the thought process you can refer: https://hetarth02.hashnode.dev/series/gsoc + +PSEUDOCODE: + +Searching happens automatically as the user types or adjusts the selected filters. +To preserve responsiveness, as much as possible of the slow parts of the search are done +in a web worker. Searching and result generation are done in the worker, and filtering and +DOM updates are done in the main thread. The filters are in the main thread as they should +be very quick to apply. This lets filters be changed without re-searching with minisearch +(which is possible even if filtering is on the worker thread) and also lets filters be +changed _while_ the worker is searching and without message passing (neither of which are +possible if filtering is on the worker thread) + +SEARCH WORKER: + +Import minisearch + +Build index + +On message from main thread + run search + find the first 200 unique results from each category, and compute their divs for display + note that this is necessary and sufficient information for the main thread to find the + first 200 unique results from any given filter set + post results to main thread + +MAIN: + +Launch worker + +Declare nonconstant globals (worker_is_running, last_search_text, unfiltered_results) + +On text update + if worker is not running, launch_search() + +launch_search + set worker_is_running to true, set last_search_text to the search text + post the search query to worker + +on message from worker + if last_search_text is not the same as the text in the search field, + the latest search result is not reflective of the latest search query, so update again + launch_search() + otherwise + set worker_is_running to false + + regardless, display the new search results to the user + save the unfiltered_results as a global + update_search() + +on filter click + adjust the filter selection + update_search() + +update_search + apply search filters by looping through the unfiltered_results and finding the first 200 + unique results that match the filters + + Update the DOM +*/ + +/////// SEARCH WORKER /////// + +function worker_function(documenterSearchIndex, documenterBaseURL, filters) { + importScripts( + "https://cdn.jsdelivr.net/npm/minisearch@6.1.0/dist/umd/index.min.js" + ); + + let data = documenterSearchIndex.map((x, key) => { + x["id"] = key; // minisearch requires a unique for each object + return x; + }); + + // list below is the lunr 2.1.3 list minus the intersect with names(Base) + // (all, any, get, in, is, only, which) and (do, else, for, let, where, while, with) + // ideally we'd just filter the original list but it's not available as a variable + const stopWords = new Set([ + "a", + "able", + "about", + "across", + "after", + "almost", + "also", + "am", + "among", + "an", + "and", + "are", + "as", + "at", + "be", + "because", + "been", + "but", + "by", + "can", + "cannot", + "could", + "dear", + "did", + "does", + "either", + "ever", + "every", + "from", + "got", + "had", + "has", + "have", + "he", + "her", + "hers", + "him", + "his", + "how", + "however", + "i", + "if", + "into", + "it", + "its", + "just", + "least", + "like", + "likely", + "may", + "me", + "might", + "most", + "must", + "my", + "neither", + "no", + "nor", + "not", + "of", + "off", + "often", + "on", + "or", + "other", + "our", + "own", + "rather", + "said", + "say", + "says", + "she", + "should", + "since", + "so", + "some", + "than", + "that", + "the", + "their", + "them", + "then", + "there", + "these", + "they", + "this", + "tis", + "to", + "too", + "twas", + "us", + "wants", + "was", + "we", + "were", + "what", + "when", + "who", + "whom", + "why", + "will", + "would", + "yet", + "you", + "your", + ]); + + let index = new MiniSearch({ + fields: ["title", "text"], // fields to index for full-text search + storeFields: ["location", "title", "text", "category", "page"], // fields to return with results + processTerm: (term) => { + let word = stopWords.has(term) ? null : term; + if (word) { + // custom trimmer that doesn't strip @ and !, which are used in julia macro and function names + word = word + .replace(/^[^a-zA-Z0-9@!]+/, "") + .replace(/[^a-zA-Z0-9@!]+$/, ""); + + word = word.toLowerCase(); + } + + return word ?? null; + }, + // add . as a separator, because otherwise "title": "Documenter.Anchors.add!", would not + // find anything if searching for "add!", only for the entire qualification + tokenize: (string) => string.split(/[\s\-\.]+/), + // options which will be applied during the search + searchOptions: { + prefix: true, + boost: { title: 100 }, + fuzzy: 2, + }, + }); + + index.addAll(data); + + /** + * Used to map characters to HTML entities. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const htmlEscapes = { + "&": "&", + "<": "<", + ">": ">", + '"': """, + "'": "'", + }; + + /** + * Used to match HTML entities and HTML characters. + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + const reUnescapedHtml = /[&<>"']/g; + const reHasUnescapedHtml = RegExp(reUnescapedHtml.source); + + /** + * Escape function from lodash + * Refer: https://github.com/lodash/lodash/blob/main/src/escape.ts + */ + function escape(string) { + return string && reHasUnescapedHtml.test(string) + ? string.replace(reUnescapedHtml, (chr) => htmlEscapes[chr]) + : string || ""; + } + + /** + * RegX escape function from MDN + * Refer: https://developer.mozilla.org/en-US/docs/Web/JavaScript/Guide/Regular_Expressions#escaping + */ + function escapeRegExp(string) { + return string.replace(/[.*+?^${}()|[\]\\]/g, "\\$&"); // $& means the whole matched string + } + + /** + * Make the result component given a minisearch result data object and the value + * of the search input as queryString. To view the result object structure, refer: + * https://lucaong.github.io/minisearch/modules/_minisearch_.html#searchresult + * + * @param {object} result + * @param {string} querystring + * @returns string + */ + function make_search_result(result, querystring) { + let search_divider = `
`; + let display_link = + result.location.slice(Math.max(0), Math.min(50, result.location.length)) + + (result.location.length > 30 ? "..." : ""); // To cut-off the link because it messes with the overflow of the whole div + + if (result.page !== "") { + display_link += ` (${result.page})`; + } + searchstring = escapeRegExp(querystring); + let textindex = new RegExp(`${searchstring}`, "i").exec(result.text); + let text = + textindex !== null + ? result.text.slice( + Math.max(textindex.index - 100, 0), + Math.min( + textindex.index + querystring.length + 100, + result.text.length + ) + ) + : ""; // cut-off text before and after from the match + + text = text.length ? escape(text) : ""; + + let display_result = text.length + ? "..." + + text.replace( + new RegExp(`${escape(searchstring)}`, "i"), // For first occurrence + '$&' + ) + + "..." + : ""; // highlights the match + + let in_code = false; + if (!["page", "section"].includes(result.category.toLowerCase())) { + in_code = true; + } + + // We encode the full url to escape some special characters which can lead to broken links + let result_div = ` + +
+
${escape(result.title)}
+
${result.category}
+
+

+ ${display_result} +

+
+ ${display_link} +
+ + ${search_divider} + `; + + return result_div; + } + + self.onmessage = function (e) { + let query = e.data; + let results = index.search(query, { + filter: (result) => { + // Only return relevant results + return result.score >= 1; + }, + combineWith: "AND", + }); + + // Pre-filter to deduplicate and limit to 200 per category to the extent + // possible without knowing what the filters are. + let filtered_results = []; + let counts = {}; + for (let filter of filters) { + counts[filter] = 0; + } + let present = {}; + + for (let result of results) { + cat = result.category; + cnt = counts[cat]; + if (cnt < 200) { + id = cat + "---" + result.location; + if (present[id]) { + continue; + } + present[id] = true; + filtered_results.push({ + location: result.location, + category: cat, + div: make_search_result(result, query), + }); + } + } + + postMessage(filtered_results); + }; +} + +/////// SEARCH MAIN /////// + +function runSearchMainCode() { + // `worker = Threads.@spawn worker_function(documenterSearchIndex)`, but in JavaScript! + const filters = [ + ...new Set(documenterSearchIndex["docs"].map((x) => x.category)), + ]; + const worker_str = + "(" + + worker_function.toString() + + ")(" + + JSON.stringify(documenterSearchIndex["docs"]) + + "," + + JSON.stringify(documenterBaseURL) + + "," + + JSON.stringify(filters) + + ")"; + const worker_blob = new Blob([worker_str], { type: "text/javascript" }); + const worker = new Worker(URL.createObjectURL(worker_blob)); + + // Whether the worker is currently handling a search. This is a boolean + // as the worker only ever handles 1 or 0 searches at a time. + var worker_is_running = false; + + // The last search text that was sent to the worker. This is used to determine + // if the worker should be launched again when it reports back results. + var last_search_text = ""; + + // The results of the last search. This, in combination with the state of the filters + // in the DOM, is used compute the results to display on calls to update_search. + var unfiltered_results = []; + + // Which filter is currently selected + var selected_filter = ""; + + $(document).on("input", ".documenter-search-input", function (event) { + if (!worker_is_running) { + launch_search(); + } + }); + + function launch_search() { + worker_is_running = true; + last_search_text = $(".documenter-search-input").val(); + worker.postMessage(last_search_text); + } + + worker.onmessage = function (e) { + if (last_search_text !== $(".documenter-search-input").val()) { + launch_search(); + } else { + worker_is_running = false; + } + + unfiltered_results = e.data; + update_search(); + }; + + $(document).on("click", ".search-filter", function () { + if ($(this).hasClass("search-filter-selected")) { + selected_filter = ""; + } else { + selected_filter = $(this).text().toLowerCase(); + } + + // This updates search results and toggles classes for UI: + update_search(); + }); + + /** + * Make/Update the search component + */ + function update_search() { + let querystring = $(".documenter-search-input").val(); + + if (querystring.trim()) { + if (selected_filter == "") { + results = unfiltered_results; + } else { + results = unfiltered_results.filter((result) => { + return selected_filter == result.category.toLowerCase(); + }); + } + + let search_result_container = ``; + let modal_filters = make_modal_body_filters(); + let search_divider = `
`; + + if (results.length) { + let links = []; + let count = 0; + let search_results = ""; + + for (var i = 0, n = results.length; i < n && count < 200; ++i) { + let result = results[i]; + if (result.location && !links.includes(result.location)) { + search_results += result.div; + count++; + links.push(result.location); + } + } + + if (count == 1) { + count_str = "1 result"; + } else if (count == 200) { + count_str = "200+ results"; + } else { + count_str = count + " results"; + } + let result_count = `
${count_str}
`; + + search_result_container = ` +
+ ${modal_filters} + ${search_divider} + ${result_count} +
+ ${search_results} +
+
+ `; + } else { + search_result_container = ` +
+ ${modal_filters} + ${search_divider} +
0 result(s)
+
+
No result found!
+ `; + } + + if ($(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").removeClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(search_result_container); + } else { + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".search-modal-card-body").html(` +
Type something to get started!
+ `); + } + } + + /** + * Make the modal filter html + * + * @returns string + */ + function make_modal_body_filters() { + let str = filters + .map((val) => { + if (selected_filter == val.toLowerCase()) { + return `${val}`; + } else { + return `${val}`; + } + }) + .join(""); + + return ` +
+ Filters: + ${str} +
`; + } +} + +function waitUntilSearchIndexAvailable() { + // It is possible that the documenter.js script runs before the page + // has finished loading and documenterSearchIndex gets defined. + // So we need to wait until the search index actually loads before setting + // up all the search-related stuff. + if (typeof documenterSearchIndex !== "undefined") { + runSearchMainCode(); + } else { + console.warn("Search Index not available, waiting"); + setTimeout(waitUntilSearchIndexAvailable, 1000); + } +} + +// The actual entry point to the search code +waitUntilSearchIndexAvailable(); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Modal settings dialog +$(document).ready(function () { + var settings = $("#documenter-settings"); + $("#documenter-settings-button").click(function () { + settings.toggleClass("is-active"); + }); + // Close the dialog if X is clicked + $("#documenter-settings button.delete").click(function () { + settings.removeClass("is-active"); + }); + // Close dialog if ESC is pressed + $(document).keyup(function (e) { + if (e.keyCode == 27) settings.removeClass("is-active"); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +$(document).ready(function () { + let search_modal_header = ` +
+
+

+ + + + +

+
+
+ +
+
+ `; + + let initial_search_body = ` +
Type something to get started!
+ `; + + let search_modal_footer = ` + + `; + + $(document.body).append( + ` + + ` + ); + + document.querySelector(".docs-search-query").addEventListener("click", () => { + openModal(); + }); + + document + .querySelector(".close-search-modal") + .addEventListener("click", () => { + closeModal(); + }); + + $(document).on("click", ".search-result-link", function () { + closeModal(); + }); + + document.addEventListener("keydown", (event) => { + if ((event.ctrlKey || event.metaKey) && event.key === "/") { + openModal(); + } else if (event.key === "Escape") { + closeModal(); + } + + return false; + }); + + // Functions to open and close a modal + function openModal() { + let searchModal = document.querySelector("#search-modal"); + + searchModal.classList.add("is-active"); + document.querySelector(".documenter-search-input").focus(); + } + + function closeModal() { + let searchModal = document.querySelector("#search-modal"); + let initial_search_body = ` +
Type something to get started!
+ `; + + searchModal.classList.remove("is-active"); + document.querySelector(".documenter-search-input").blur(); + + if (!$(".search-modal-card-body").hasClass("is-justify-content-center")) { + $(".search-modal-card-body").addClass("is-justify-content-center"); + } + + $(".documenter-search-input").val(""); + $(".search-modal-card-body").html(initial_search_body); + } + + document + .querySelector("#search-modal .modal-background") + .addEventListener("click", () => { + closeModal(); + }); +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Manages the showing and hiding of the sidebar. +$(document).ready(function () { + var sidebar = $("#documenter > .docs-sidebar"); + var sidebar_button = $("#documenter-sidebar-button"); + sidebar_button.click(function (ev) { + ev.preventDefault(); + sidebar.toggleClass("visible"); + if (sidebar.hasClass("visible")) { + // Makes sure that the current menu item is visible in the sidebar. + $("#documenter .docs-menu a.is-active").focus(); + } + }); + $("#documenter > .docs-main").bind("click", function (ev) { + if ($(ev.target).is(sidebar_button)) { + return; + } + if (sidebar.hasClass("visible")) { + sidebar.removeClass("visible"); + } + }); +}); + +// Resizes the package name / sitename in the sidebar if it is too wide. +// Inspired by: https://github.com/davatron5000/FitText.js +$(document).ready(function () { + e = $("#documenter .docs-autofit"); + function resize() { + var L = parseInt(e.css("max-width"), 10); + var L0 = e.width(); + if (L0 > L) { + var h0 = parseInt(e.css("font-size"), 10); + e.css("font-size", (L * h0) / L0); + // TODO: make sure it survives resizes? + } + } + // call once and then register events + resize(); + $(window).resize(resize); + $(window).on("orientationchange", resize); +}); + +// Scroll the navigation bar to the currently selected menu item +$(document).ready(function () { + var sidebar = $("#documenter .docs-menu").get(0); + var active = $("#documenter .docs-menu .is-active").get(0); + if (typeof active !== "undefined") { + sidebar.scrollTop = active.offsetTop - sidebar.offsetTop - 15; + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// Theme picker setup +$(document).ready(function () { + // onchange callback + $("#documenter-themepicker").change(function themepick_callback(ev) { + var themename = $("#documenter-themepicker option:selected").attr("value"); + if (themename === "auto") { + // set_theme(window.matchMedia('(prefers-color-scheme: dark)').matches ? 'dark' : 'light'); + window.localStorage.removeItem("documenter-theme"); + } else { + // set_theme(themename); + window.localStorage.setItem("documenter-theme", themename); + } + // We re-use the global function from themeswap.js to actually do the swapping. + set_theme_from_local_storage(); + }); + + // Make sure that the themepicker displays the correct theme when the theme is retrieved + // from localStorage + if (typeof window.localStorage !== "undefined") { + var theme = window.localStorage.getItem("documenter-theme"); + if (theme !== null) { + $("#documenter-themepicker option").each(function (i, e) { + e.selected = e.value === theme; + }); + } + } +}); + +}) +//////////////////////////////////////////////////////////////////////////////// +require(['jquery'], function($) { + +// update the version selector with info from the siteinfo.js and ../versions.js files +$(document).ready(function () { + // If the version selector is disabled with DOCUMENTER_VERSION_SELECTOR_DISABLED in the + // siteinfo.js file, we just return immediately and not display the version selector. + if ( + typeof DOCUMENTER_VERSION_SELECTOR_DISABLED === "boolean" && + DOCUMENTER_VERSION_SELECTOR_DISABLED + ) { + return; + } + + var version_selector = $("#documenter .docs-version-selector"); + var version_selector_select = $("#documenter .docs-version-selector select"); + + version_selector_select.change(function (x) { + target_href = version_selector_select + .children("option:selected") + .get(0).value; + window.location.href = target_href; + }); + + // add the current version to the selector based on siteinfo.js, but only if the selector is empty + if ( + typeof DOCUMENTER_CURRENT_VERSION !== "undefined" && + $("#version-selector > option").length == 0 + ) { + var option = $( + "" + ); + version_selector_select.append(option); + } + + if (typeof DOC_VERSIONS !== "undefined") { + var existing_versions = version_selector_select.children("option"); + var existing_versions_texts = existing_versions.map(function (i, x) { + return x.text; + }); + DOC_VERSIONS.forEach(function (each) { + var version_url = documenterBaseURL + "/../" + each + "/"; + var existing_id = $.inArray(each, existing_versions_texts); + // if not already in the version selector, add it as a new option, + // otherwise update the old option with the URL and enable it + if (existing_id == -1) { + var option = $( + "" + ); + version_selector_select.append(option); + } else { + var option = existing_versions[existing_id]; + option.value = version_url; + option.disabled = false; + } + }); + } + + // only show the version selector if the selector has been populated + if (version_selector_select.children("option").length > 0) { + version_selector.toggleClass("visible"); + } +}); + +}) diff --git a/previews/PR184/assets/makelatexfigs b/previews/PR184/assets/makelatexfigs new file mode 100755 index 00000000..de9feb1d --- /dev/null +++ b/previews/PR184/assets/makelatexfigs @@ -0,0 +1,21 @@ +#!/bin/bash + +# Function to check if a command is available +command_exists() { + command -v "$1" >/dev/null 2>&1 +} + +# Check for latexmk +if ! command_exists latexmk; then + echo "Error: 'latexmk' is not installed or not in your PATH." + exit 1 +fi + +# Check for pdf2svg +if ! command_exists pdf2svg; then + echo "Error: 'pdf2svg' is not installed or not in your PATH." + exit 1 +fi + +latexmk -pdflua -shell-escape mathmodel.tex +latexmk -C diff --git a/previews/PR184/assets/mathmodel.svg b/previews/PR184/assets/mathmodel.svg new file mode 100644 index 00000000..68e731b8 --- /dev/null +++ b/previews/PR184/assets/mathmodel.svg @@ -0,0 +1,543 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/assets/mathmodel.tex b/previews/PR184/assets/mathmodel.tex new file mode 100644 index 00000000..525421c7 --- /dev/null +++ b/previews/PR184/assets/mathmodel.tex @@ -0,0 +1,65 @@ +% \documentclass[tikz,convert={outfile=\jobname.svg}]{standalone} +\documentclass[tikz,convert=pdf2svg]{standalone} +%\usetikzlibrary{...}% tikz package already loaded by 'tikz' option +\usetikzlibrary{arrows, arrows.meta} +% \tikzset{ +% >=stealth', +% } +\usepackage{amsmath,amssymb} +\usepackage{tikz} +\usepackage{xcolor} + +\makeatletter +\def\mathcolor#1#{\@mathcolor{#1}} +\def\@mathcolor#1#2#3{% + \protect\leavevmode + \begingroup + \color#1{#2}#3% + \endgroup +} +\makeatother + +\definecolor{wblue}{HTML}{0072b2} +\definecolor{worange}{HTML}{e69f00} +\definecolor{wbblue}{HTML}{56b4e9} +\definecolor{wgreen}{HTML}{009e73} +\definecolor{wyellow}{HTML}{f0e442} + +\begin{document} +\begin{tikzpicture} + \node[draw, circle](n1){1}; + \node[draw, circle] at (10cm, 1cm)(n2){2}; + \node[draw, circle] at (2.5cm, -1.5cm)(n3){3}; + \path (n1)--(n2) coordinate[pos=.5](mid12); + \draw[-{Stealth[length=3mm, width=2mm]}] (n1)--(mid12); + \draw(mid12)--(n2); + \draw (n2)--(n3); + \draw (n3)--(n1); + \draw[] (n3) --++(-2,-1); + \draw[] (n3) --++(1,-1); + \draw[] (n1) --++(-1,-2.5); + \draw[] (n2) --++(-2,-3.5); + + \node[text width=5cm, xshift=-0.5cm, yshift=1.25cm] at (n1){% + \small\begin{align*} + M\,\dot{x} &= f^{\mathrm{v}}_{1}\left(x, \sum_{k} \mathcolor{worange}{e_{k1}}, p, t\right)\\ + \mathcolor{worange}{v_{1}} &= g^{\mathrm{v}}_{1}\left(x, p, t\right)\\ + \end{align*}% + }; + \node[text width=5cm, xshift=0.5cm, yshift=1.25cm] at (n2)(block){% + \small\begin{align*} + M\,\dot{x} &= f^{\mathrm{v}}_{2}\left(x, \sum_{k} \mathcolor{wblue}{e_{k2}}, p_{\mathrm{v},2}, t\right)\\ + \mathcolor{wblue}{v_{2}} &= g^{\mathrm{v}}_{2}\left(x, p_{\mathrm{v},2}, t\right)\\ + \end{align*}% + }; + + \path (n1)--(n2) node[pos=.5,text width=5cm, xshift=0cm, yshift=1.25cm]{% + \small\begin{align*} + M\,\dot{x} &= f^{\mathrm{e}}_{12}\left(x, \mathcolor{worange}{v_{\mathrm{src}}}, \mathcolor{wblue}{v_{\mathrm{dst}}}, p, t\right)\\ + \mathcolor{worange}{e_{21}} &= g^{\mathrm{e}}_{\mathrm{src}}\left(x, \mathcolor{worange}{v_{\mathrm{src}}}, \mathcolor{wblue}{v_{\mathrm{dst}}}, p, t\right)\\ + \mathcolor{wblue}{e_{12}} &= g^{\mathrm{e}}_{\mathrm{dst}}\left(x, \mathcolor{worange}{v_{\mathrm{src}}}, \mathcolor{wblue}{v_{\mathrm{dst}}}, p, t\right)\\ + \end{align*}% + }; + +\end{tikzpicture} +\end{document} diff --git a/previews/PR184/assets/themes/catppuccin-frappe.css b/previews/PR184/assets/themes/catppuccin-frappe.css new file mode 100644 index 00000000..32e3f008 --- /dev/null +++ b/previews/PR184/assets/themes/catppuccin-frappe.css @@ -0,0 +1 @@ +html.theme--catppuccin-frappe .pagination-previous,html.theme--catppuccin-frappe .pagination-next,html.theme--catppuccin-frappe .pagination-link,html.theme--catppuccin-frappe .pagination-ellipsis,html.theme--catppuccin-frappe .file-cta,html.theme--catppuccin-frappe .file-name,html.theme--catppuccin-frappe .select select,html.theme--catppuccin-frappe .textarea,html.theme--catppuccin-frappe .input,html.theme--catppuccin-frappe #documenter .docs-sidebar 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html.theme--catppuccin-frappe .content kbd.button.is-outlined{background-color:transparent;border-color:#414559;box-shadow:none;color:#414559}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-frappe .content 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.button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-link.is-outlined{background-color:transparent;border-color:#8caaee;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-outlined:hover,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-link.is-outlined:focus,html.theme--catppuccin-frappe .button.is-link.is-outlined.is-focused{background-color:#8caaee;border-color:#8caaee;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe 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.button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#8caaee}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8caaee #8caaee !important}html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-link.is-light{background-color:#edf2fc;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:hover,html.theme--catppuccin-frappe .button.is-link.is-light.is-hovered{background-color:#e2eafb;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-link.is-light:active,html.theme--catppuccin-frappe .button.is-link.is-light.is-active{background-color:#d7e1f9;border-color:transparent;color:#153a8e}html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:hover,html.theme--catppuccin-frappe .button.is-info.is-hovered{background-color:#78c4b9;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus,html.theme--catppuccin-frappe .button.is-info.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info:focus:not(:active),html.theme--catppuccin-frappe .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(129,200,190,0.25)}html.theme--catppuccin-frappe .button.is-info:active,html.theme--catppuccin-frappe .button.is-info.is-active{background-color:#6fc0b5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info{background-color:#81c8be;border-color:#81c8be;box-shadow:none}html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-focused{background-color:#81c8be;border-color:#81c8be;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-outlined{background-color:transparent;border-color:#81c8be;box-shadow:none;color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#81c8be}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #81c8be #81c8be !important}html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-info.is-light{background-color:#f1f9f8;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:hover,html.theme--catppuccin-frappe .button.is-info.is-light.is-hovered{background-color:#e8f5f3;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe .button.is-info.is-light:active,html.theme--catppuccin-frappe .button.is-info.is-light.is-active{background-color:#dff1ef;border-color:transparent;color:#2d675f}html.theme--catppuccin-frappe 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.button.is-success{background-color:#a6d189;border-color:#a6d189;box-shadow:none}html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-focused{background-color:#a6d189;border-color:#a6d189;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-outlined{background-color:transparent;border-color:#a6d189;box-shadow:none;color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6d189}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6d189 #a6d189 !important}html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-success.is-light{background-color:#f4f9f0;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:hover,html.theme--catppuccin-frappe .button.is-success.is-light.is-hovered{background-color:#edf6e7;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-success.is-light:active,html.theme--catppuccin-frappe .button.is-success.is-light.is-active{background-color:#e6f2de;border-color:transparent;color:#446a29}html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:hover,html.theme--catppuccin-frappe .button.is-warning.is-hovered{background-color:#e3c386;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus,html.theme--catppuccin-frappe .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning:focus:not(:active),html.theme--catppuccin-frappe .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(229,200,144,0.25)}html.theme--catppuccin-frappe .button.is-warning:active,html.theme--catppuccin-frappe .button.is-warning.is-active{background-color:#e0be7b;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning{background-color:#e5c890;border-color:#e5c890;box-shadow:none}html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-frappe .button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-outlined:hover,html.theme--catppuccin-frappe 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.button.is-warning.is-outlined{background-color:transparent;border-color:#e5c890;box-shadow:none;color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#e5c890}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e5c890 #e5c890 !important}html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-frappe .button.is-warning.is-light{background-color:#fbf7ee;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:hover,html.theme--catppuccin-frappe .button.is-warning.is-light.is-hovered{background-color:#f9f2e4;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-warning.is-light:active,html.theme--catppuccin-frappe .button.is-warning.is-light.is-active{background-color:#f6edda;border-color:transparent;color:#78591c}html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:hover,html.theme--catppuccin-frappe .button.is-danger.is-hovered{background-color:#e57779;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus,html.theme--catppuccin-frappe .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger:focus:not(:active),html.theme--catppuccin-frappe .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(231,130,132,0.25)}html.theme--catppuccin-frappe .button.is-danger:active,html.theme--catppuccin-frappe .button.is-danger.is-active{background-color:#e36d6f;border-color:transparent;color:#fff}html.theme--catppuccin-frappe .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger{background-color:#e78284;border-color:#e78284;box-shadow:none}html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-frappe .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-focused{background-color:#e78284;border-color:#e78284;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-frappe .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-outlined{background-color:transparent;border-color:#e78284;box-shadow:none;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#e78284}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #e78284 #e78284 !important}html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-frappe .button.is-danger.is-light{background-color:#fceeee;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:hover,html.theme--catppuccin-frappe .button.is-danger.is-light.is-hovered{background-color:#fae3e4;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-danger.is-light:active,html.theme--catppuccin-frappe .button.is-danger.is-light.is-active{background-color:#f8d8d9;border-color:transparent;color:#9a1e20}html.theme--catppuccin-frappe .button.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-frappe .button.is-small:not(.is-rounded),html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .button.is-normal{font-size:1rem}html.theme--catppuccin-frappe .button.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .button.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .button[disabled],fieldset[disabled] html.theme--catppuccin-frappe .button{background-color:#737994;border-color:#626880;box-shadow:none;opacity:.5}html.theme--catppuccin-frappe .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-frappe .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-frappe .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-frappe .button.is-static{background-color:#292c3c;border-color:#626880;color:#838ba7;box-shadow:none;pointer-events:none}html.theme--catppuccin-frappe .button.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-frappe .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-frappe .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-frappe .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-frappe .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-frappe .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-frappe .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-frappe .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-frappe .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-frappe .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-frappe .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-frappe .buttons.has-addons .button:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-frappe .buttons.has-addons .button:focus,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused,html.theme--catppuccin-frappe .buttons.has-addons .button:active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-frappe .buttons.has-addons .button:focus:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-frappe .buttons.has-addons .button:active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-frappe .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-frappe .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-frappe .buttons.is-centered{justify-content:center}html.theme--catppuccin-frappe .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-frappe .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-frappe .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-frappe .button.is-responsive.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-frappe .button.is-responsive,html.theme--catppuccin-frappe .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-frappe .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-frappe .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-frappe .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-frappe .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-frappe .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-frappe .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-frappe .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-frappe .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-frappe .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-frappe .content li+li{margin-top:0.25em}html.theme--catppuccin-frappe .content p:not(:last-child),html.theme--catppuccin-frappe .content dl:not(:last-child),html.theme--catppuccin-frappe .content ol:not(:last-child),html.theme--catppuccin-frappe .content ul:not(:last-child),html.theme--catppuccin-frappe .content blockquote:not(:last-child),html.theme--catppuccin-frappe .content pre:not(:last-child),html.theme--catppuccin-frappe .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-frappe .content h1,html.theme--catppuccin-frappe .content h2,html.theme--catppuccin-frappe .content h3,html.theme--catppuccin-frappe .content h4,html.theme--catppuccin-frappe .content h5,html.theme--catppuccin-frappe .content h6{color:#c6d0f5;font-weight:600;line-height:1.125}html.theme--catppuccin-frappe .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-frappe .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-frappe .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-frappe .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-frappe .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-frappe .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-frappe .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-frappe .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-frappe .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-frappe .content blockquote{background-color:#292c3c;border-left:5px solid #626880;padding:1.25em 1.5em}html.theme--catppuccin-frappe .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-frappe .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-frappe .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-frappe .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-frappe .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-frappe .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-frappe .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-frappe .content ul ul ul{list-style-type:square}html.theme--catppuccin-frappe .content dd{margin-left:2em}html.theme--catppuccin-frappe .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-frappe .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-frappe .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-frappe .content figure img{display:inline-block}html.theme--catppuccin-frappe .content figure figcaption{font-style:italic}html.theme--catppuccin-frappe .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-frappe .content sup,html.theme--catppuccin-frappe .content sub{font-size:75%}html.theme--catppuccin-frappe .content table{width:100%}html.theme--catppuccin-frappe .content table td,html.theme--catppuccin-frappe .content table th{border:1px solid #626880;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-frappe .content table th{color:#b0bef1}html.theme--catppuccin-frappe .content table th:not([align]){text-align:inherit}html.theme--catppuccin-frappe .content table thead td,html.theme--catppuccin-frappe .content table thead th{border-width:0 0 2px;color:#b0bef1}html.theme--catppuccin-frappe .content table tfoot td,html.theme--catppuccin-frappe .content table tfoot th{border-width:2px 0 0;color:#b0bef1}html.theme--catppuccin-frappe .content table tbody tr:last-child td,html.theme--catppuccin-frappe .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-frappe .content .tabs li+li{margin-top:0}html.theme--catppuccin-frappe .content.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-frappe .content.is-normal{font-size:1rem}html.theme--catppuccin-frappe .content.is-medium{font-size:1.25rem}html.theme--catppuccin-frappe .content.is-large{font-size:1.5rem}html.theme--catppuccin-frappe .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-frappe .icon.is-small,html.theme--catppuccin-frappe #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-frappe .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-frappe .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-frappe .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-frappe .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-frappe .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-frappe .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-frappe div.icon-text{display:flex}html.theme--catppuccin-frappe .image,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-frappe .image img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-frappe .image img.is-rounded,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-frappe .image.is-fullwidth,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-frappe .image.is-square img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-frappe .image.is-square .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-frappe .image.is-1by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-frappe .image.is-1by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-frappe .image.is-5by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-frappe .image.is-5by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-frappe .image.is-4by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-frappe .image.is-4by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-frappe .image.is-3by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-frappe .image.is-5by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-frappe .image.is-5by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-frappe .image.is-16by9 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-frappe .image.is-16by9 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-frappe .image.is-2by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-frappe .image.is-2by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-frappe .image.is-3by1 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-frappe .image.is-3by1 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by1 .has-ratio,html.theme--catppuccin-frappe .image.is-4by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 img,html.theme--catppuccin-frappe .image.is-4by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by5 .has-ratio,html.theme--catppuccin-frappe .image.is-3by4 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 img,html.theme--catppuccin-frappe .image.is-3by4 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by4 .has-ratio,html.theme--catppuccin-frappe .image.is-2by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 img,html.theme--catppuccin-frappe .image.is-2by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-2by3 .has-ratio,html.theme--catppuccin-frappe .image.is-3by5 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-frappe .image.is-3by5 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-frappe .image.is-9by16 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-frappe .image.is-9by16 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-frappe .image.is-1by2 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-frappe .image.is-1by2 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-frappe .image.is-1by3 img,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-frappe .image.is-1by3 .has-ratio,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by3 .has-ratio{height:100%;width:100%}html.theme--catppuccin-frappe .image.is-square,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-square,html.theme--catppuccin-frappe .image.is-1by1,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-1by1{padding-top:100%}html.theme--catppuccin-frappe .image.is-5by4,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-5by4{padding-top:80%}html.theme--catppuccin-frappe .image.is-4by3,html.theme--catppuccin-frappe #documenter .docs-sidebar .docs-logo>img.is-4by3{padding-top:75%}html.theme--catppuccin-frappe 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.modal-card{display:flex;flex-direction:column;max-height:calc(100vh - 40px);overflow:hidden;-ms-overflow-y:visible}html.theme--catppuccin-frappe .modal-card-head,html.theme--catppuccin-frappe .modal-card-foot{align-items:center;background-color:#292c3c;display:flex;flex-shrink:0;justify-content:flex-start;padding:20px;position:relative}html.theme--catppuccin-frappe .modal-card-head{border-bottom:1px solid #626880;border-top-left-radius:8px;border-top-right-radius:8px}html.theme--catppuccin-frappe .modal-card-title{color:#c6d0f5;flex-grow:1;flex-shrink:0;font-size:1.5rem;line-height:1}html.theme--catppuccin-frappe .modal-card-foot{border-bottom-left-radius:8px;border-bottom-right-radius:8px;border-top:1px solid #626880}html.theme--catppuccin-frappe .modal-card-foot .button:not(:last-child){margin-right:.5em}html.theme--catppuccin-frappe 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kbd.button.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#ccd0da}html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ccd0da #ccd0da !important}html.theme--catppuccin-latte 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.docstring>section>a.button.is-inverted.is-outlined.is-loading.docs-sourcelink:focus::after,html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.is-loading.is-focused.docs-sourcelink::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined[disabled],html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-primary.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-latte .docstring>section>a.button.is-inverted.is-outlined.docs-sourcelink{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-primary.is-light,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:hover,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:hover,html.theme--catppuccin-latte .button.is-primary.is-light.is-hovered,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-hovered.docs-sourcelink{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-primary.is-light:active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.docs-sourcelink:active,html.theme--catppuccin-latte .button.is-primary.is-light.is-active,html.theme--catppuccin-latte .docstring>section>a.button.is-light.is-active.docs-sourcelink{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:hover,html.theme--catppuccin-latte .button.is-link.is-hovered{background-color:#125ef4;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus,html.theme--catppuccin-latte .button.is-link.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link:focus:not(:active),html.theme--catppuccin-latte .button.is-link.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(30,102,245,0.25)}html.theme--catppuccin-latte .button.is-link:active,html.theme--catppuccin-latte .button.is-link.is-active{background-color:#0b57ef;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-link[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link{background-color:#1e66f5;border-color:#1e66f5;box-shadow:none}html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-link.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-outlined.is-focused{background-color:#1e66f5;border-color:#1e66f5;color:#fff}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-outlined{background-color:transparent;border-color:#1e66f5;box-shadow:none;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1e66f5}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1e66f5 #1e66f5 !important}html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-link.is-light{background-color:#ebf2fe;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:hover,html.theme--catppuccin-latte .button.is-link.is-light.is-hovered{background-color:#dfe9fe;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-link.is-light:active,html.theme--catppuccin-latte .button.is-link.is-light.is-active{background-color:#d3e1fd;border-color:transparent;color:#0a52e1}html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:hover,html.theme--catppuccin-latte .button.is-info.is-hovered{background-color:#15878e;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus,html.theme--catppuccin-latte .button.is-info.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info:focus:not(:active),html.theme--catppuccin-latte .button.is-info.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(23,146,153,0.25)}html.theme--catppuccin-latte .button.is-info:active,html.theme--catppuccin-latte .button.is-info.is-active{background-color:#147d83;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-info[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info{background-color:#179299;border-color:#179299;box-shadow:none}html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-info.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;color:#179299}html.theme--catppuccin-latte .button.is-info.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-outlined.is-focused{background-color:#179299;border-color:#179299;color:#fff}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-outlined{background-color:transparent;border-color:#179299;box-shadow:none;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#179299}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #179299 #179299 !important}html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-info.is-light{background-color:#edfcfc;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:hover,html.theme--catppuccin-latte .button.is-info.is-light.is-hovered{background-color:#e2f9fb;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-info.is-light:active,html.theme--catppuccin-latte .button.is-info.is-light.is-active{background-color:#d7f7f9;border-color:transparent;color:#1cb2ba}html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:hover,html.theme--catppuccin-latte .button.is-success.is-hovered{background-color:#3c9628;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus,html.theme--catppuccin-latte .button.is-success.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success:focus:not(:active),html.theme--catppuccin-latte .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(64,160,43,0.25)}html.theme--catppuccin-latte .button.is-success:active,html.theme--catppuccin-latte .button.is-success.is-active{background-color:#388c26;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success{background-color:#40a02b;border-color:#40a02b;box-shadow:none}html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-outlined.is-focused{background-color:#40a02b;border-color:#40a02b;color:#fff}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-outlined{background-color:transparent;border-color:#40a02b;box-shadow:none;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#40a02b}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #40a02b #40a02b !important}html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-success.is-light{background-color:#f1fbef;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:hover,html.theme--catppuccin-latte .button.is-success.is-light.is-hovered{background-color:#e8f8e5;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-success.is-light:active,html.theme--catppuccin-latte .button.is-success.is-light.is-active{background-color:#e0f5db;border-color:transparent;color:#40a12b}html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:hover,html.theme--catppuccin-latte .button.is-warning.is-hovered{background-color:#d4871c;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus,html.theme--catppuccin-latte .button.is-warning.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning:focus:not(:active),html.theme--catppuccin-latte .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(223,142,29,0.25)}html.theme--catppuccin-latte .button.is-warning:active,html.theme--catppuccin-latte .button.is-warning.is-active{background-color:#c8801a;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning{background-color:#df8e1d;border-color:#df8e1d;box-shadow:none}html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-focused{background-color:#df8e1d;border-color:#df8e1d;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-outlined{background-color:transparent;border-color:#df8e1d;box-shadow:none;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-focused{background-color:#fff;color:#df8e1d}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #df8e1d #df8e1d !important}html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-warning.is-light{background-color:#fdf6ed;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:hover,html.theme--catppuccin-latte .button.is-warning.is-light.is-hovered{background-color:#fbf1e2;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-warning.is-light:active,html.theme--catppuccin-latte .button.is-warning.is-light.is-active{background-color:#faebd6;border-color:transparent;color:#9e6515}html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:hover,html.theme--catppuccin-latte .button.is-danger.is-hovered{background-color:#c60e36;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus,html.theme--catppuccin-latte .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger:focus:not(:active),html.theme--catppuccin-latte .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(210,15,57,0.25)}html.theme--catppuccin-latte .button.is-danger:active,html.theme--catppuccin-latte .button.is-danger.is-active{background-color:#ba0d33;border-color:transparent;color:#fff}html.theme--catppuccin-latte .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger{background-color:#d20f39;border-color:#d20f39;box-shadow:none}html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-latte .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-focused{background-color:#d20f39;border-color:#d20f39;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-latte .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-outlined{background-color:transparent;border-color:#d20f39;box-shadow:none;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#d20f39}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #d20f39 #d20f39 !important}html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-latte .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-latte .button.is-danger.is-light{background-color:#feecf0;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:hover,html.theme--catppuccin-latte .button.is-danger.is-light.is-hovered{background-color:#fde0e6;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-danger.is-light:active,html.theme--catppuccin-latte .button.is-danger.is-light.is-active{background-color:#fcd4dd;border-color:transparent;color:#e9113f}html.theme--catppuccin-latte .button.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-latte .button.is-small:not(.is-rounded),html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .button.is-normal{font-size:1rem}html.theme--catppuccin-latte .button.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .button.is-large{font-size:1.5rem}html.theme--catppuccin-latte .button[disabled],fieldset[disabled] html.theme--catppuccin-latte .button{background-color:#9ca0b0;border-color:#acb0be;box-shadow:none;opacity:.5}html.theme--catppuccin-latte .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-latte .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-latte .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-latte .button.is-static{background-color:#e6e9ef;border-color:#acb0be;color:#8c8fa1;box-shadow:none;pointer-events:none}html.theme--catppuccin-latte .button.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-latte .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-latte .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-latte .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-latte .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-latte .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-latte .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-latte .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-latte .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-latte .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-latte .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-latte .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-latte .buttons.has-addons .button:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-latte .buttons.has-addons .button:focus,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused,html.theme--catppuccin-latte .buttons.has-addons .button:active,html.theme--catppuccin-latte .buttons.has-addons .button.is-active,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-latte .buttons.has-addons .button:focus:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-latte .buttons.has-addons .button:active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-latte .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-latte .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-latte .buttons.is-centered{justify-content:center}html.theme--catppuccin-latte .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-latte .buttons.is-right{justify-content:flex-end}html.theme--catppuccin-latte .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--catppuccin-latte .button.is-responsive.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--catppuccin-latte .button.is-responsive,html.theme--catppuccin-latte .button.is-responsive.is-normal{font-size:.75rem}html.theme--catppuccin-latte .button.is-responsive.is-medium{font-size:1rem}html.theme--catppuccin-latte .button.is-responsive.is-large{font-size:1.25rem}}html.theme--catppuccin-latte .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--catppuccin-latte .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--catppuccin-latte .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--catppuccin-latte .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--catppuccin-latte .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--catppuccin-latte .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--catppuccin-latte .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--catppuccin-latte .content li+li{margin-top:0.25em}html.theme--catppuccin-latte .content p:not(:last-child),html.theme--catppuccin-latte .content dl:not(:last-child),html.theme--catppuccin-latte .content ol:not(:last-child),html.theme--catppuccin-latte .content ul:not(:last-child),html.theme--catppuccin-latte .content blockquote:not(:last-child),html.theme--catppuccin-latte .content pre:not(:last-child),html.theme--catppuccin-latte .content table:not(:last-child){margin-bottom:1em}html.theme--catppuccin-latte .content h1,html.theme--catppuccin-latte .content h2,html.theme--catppuccin-latte .content h3,html.theme--catppuccin-latte .content h4,html.theme--catppuccin-latte .content h5,html.theme--catppuccin-latte .content h6{color:#4c4f69;font-weight:600;line-height:1.125}html.theme--catppuccin-latte .content h1{font-size:2em;margin-bottom:0.5em}html.theme--catppuccin-latte .content h1:not(:first-child){margin-top:1em}html.theme--catppuccin-latte .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--catppuccin-latte .content h2:not(:first-child){margin-top:1.1428em}html.theme--catppuccin-latte .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--catppuccin-latte .content h3:not(:first-child){margin-top:1.3333em}html.theme--catppuccin-latte .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--catppuccin-latte .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--catppuccin-latte .content h6{font-size:1em;margin-bottom:1em}html.theme--catppuccin-latte .content blockquote{background-color:#e6e9ef;border-left:5px solid #acb0be;padding:1.25em 1.5em}html.theme--catppuccin-latte .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ol:not([type]){list-style-type:decimal}html.theme--catppuccin-latte .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--catppuccin-latte .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--catppuccin-latte .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--catppuccin-latte .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--catppuccin-latte .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--catppuccin-latte .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--catppuccin-latte .content ul ul ul{list-style-type:square}html.theme--catppuccin-latte .content dd{margin-left:2em}html.theme--catppuccin-latte .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--catppuccin-latte .content figure:not(:first-child){margin-top:2em}html.theme--catppuccin-latte .content figure:not(:last-child){margin-bottom:2em}html.theme--catppuccin-latte .content figure img{display:inline-block}html.theme--catppuccin-latte .content figure figcaption{font-style:italic}html.theme--catppuccin-latte .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--catppuccin-latte .content sup,html.theme--catppuccin-latte .content sub{font-size:75%}html.theme--catppuccin-latte .content table{width:100%}html.theme--catppuccin-latte .content table td,html.theme--catppuccin-latte .content table th{border:1px solid #acb0be;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--catppuccin-latte .content table th{color:#41445a}html.theme--catppuccin-latte .content table th:not([align]){text-align:inherit}html.theme--catppuccin-latte .content table thead td,html.theme--catppuccin-latte .content table thead th{border-width:0 0 2px;color:#41445a}html.theme--catppuccin-latte .content table tfoot td,html.theme--catppuccin-latte .content table tfoot th{border-width:2px 0 0;color:#41445a}html.theme--catppuccin-latte .content table tbody tr:last-child td,html.theme--catppuccin-latte .content table tbody tr:last-child th{border-bottom-width:0}html.theme--catppuccin-latte .content .tabs li+li{margin-top:0}html.theme--catppuccin-latte .content.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--catppuccin-latte .content.is-normal{font-size:1rem}html.theme--catppuccin-latte .content.is-medium{font-size:1.25rem}html.theme--catppuccin-latte .content.is-large{font-size:1.5rem}html.theme--catppuccin-latte .icon{align-items:center;display:inline-flex;justify-content:center;height:1.5rem;width:1.5rem}html.theme--catppuccin-latte .icon.is-small,html.theme--catppuccin-latte #documenter .docs-sidebar form.docs-search>input.icon{height:1rem;width:1rem}html.theme--catppuccin-latte .icon.is-medium{height:2rem;width:2rem}html.theme--catppuccin-latte .icon.is-large{height:3rem;width:3rem}html.theme--catppuccin-latte .icon-text{align-items:flex-start;color:inherit;display:inline-flex;flex-wrap:wrap;line-height:1.5rem;vertical-align:top}html.theme--catppuccin-latte .icon-text .icon{flex-grow:0;flex-shrink:0}html.theme--catppuccin-latte .icon-text .icon:not(:last-child){margin-right:.25em}html.theme--catppuccin-latte .icon-text .icon:not(:first-child){margin-left:.25em}html.theme--catppuccin-latte div.icon-text{display:flex}html.theme--catppuccin-latte .image,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img{display:block;position:relative}html.theme--catppuccin-latte .image img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--catppuccin-latte .image img.is-rounded,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--catppuccin-latte .image.is-fullwidth,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--catppuccin-latte .image.is-square img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--catppuccin-latte .image.is-square .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--catppuccin-latte .image.is-1by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--catppuccin-latte .image.is-1by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--catppuccin-latte .image.is-5by4 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 img,html.theme--catppuccin-latte .image.is-5by4 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by4 .has-ratio,html.theme--catppuccin-latte .image.is-4by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 img,html.theme--catppuccin-latte .image.is-4by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-4by3 .has-ratio,html.theme--catppuccin-latte .image.is-3by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 img,html.theme--catppuccin-latte .image.is-3by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by2 .has-ratio,html.theme--catppuccin-latte .image.is-5by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 img,html.theme--catppuccin-latte .image.is-5by3 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-5by3 .has-ratio,html.theme--catppuccin-latte .image.is-16by9 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 img,html.theme--catppuccin-latte .image.is-16by9 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-16by9 .has-ratio,html.theme--catppuccin-latte .image.is-2by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 img,html.theme--catppuccin-latte .image.is-2by1 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-2by1 .has-ratio,html.theme--catppuccin-latte .image.is-3by1 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by1 img,html.theme--catppuccin-latte 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img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 img,html.theme--catppuccin-latte .image.is-3by5 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-3by5 .has-ratio,html.theme--catppuccin-latte .image.is-9by16 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 img,html.theme--catppuccin-latte .image.is-9by16 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-9by16 .has-ratio,html.theme--catppuccin-latte .image.is-1by2 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 img,html.theme--catppuccin-latte .image.is-1by2 .has-ratio,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by2 .has-ratio,html.theme--catppuccin-latte .image.is-1by3 img,html.theme--catppuccin-latte #documenter .docs-sidebar .docs-logo>img.is-1by3 img,html.theme--catppuccin-latte .image.is-1by3 .has-ratio,html.theme--catppuccin-latte 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.button.is-white.is-inverted.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:hover,html.theme--catppuccin-macchiato .button.is-black.is-hovered{background-color:#040404;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus,html.theme--catppuccin-macchiato .button.is-black.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black:focus:not(:active),html.theme--catppuccin-macchiato .button.is-black.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(10,10,10,0.25)}html.theme--catppuccin-macchiato .button.is-black:active,html.theme--catppuccin-macchiato .button.is-black.is-active{background-color:#000;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-black[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black{background-color:#0a0a0a;border-color:#0a0a0a;box-shadow:none}html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-black.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-focused{background-color:#0a0a0a;border-color:#0a0a0a;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-black.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-outlined{background-color:transparent;border-color:#0a0a0a;box-shadow:none;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-focused{background-color:#fff;color:#0a0a0a}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #0a0a0a #0a0a0a !important}html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-black.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:hover,html.theme--catppuccin-macchiato .button.is-light.is-hovered{background-color:#eee;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus,html.theme--catppuccin-macchiato .button.is-light.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light:focus:not(:active),html.theme--catppuccin-macchiato .button.is-light.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(245,245,245,0.25)}html.theme--catppuccin-macchiato .button.is-light:active,html.theme--catppuccin-macchiato .button.is-light.is-active{background-color:#e8e8e8;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light{background-color:#f5f5f5;border-color:#f5f5f5;box-shadow:none}html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-light.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-light.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-light.is-outlined{background-color:transparent;border-color:#f5f5f5;color:#f5f5f5}html.theme--catppuccin-macchiato .button.is-light.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-light.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-focused{background-color:#f5f5f5;border-color:#f5f5f5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-light.is-outlined.is-loading::after{border-color:transparent transparent #f5f5f5 #f5f5f5 !important}html.theme--catppuccin-macchiato 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.content kbd.button{background-color:#363a4f;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:hover,html.theme--catppuccin-macchiato .content kbd.button:hover,html.theme--catppuccin-macchiato .button.is-dark.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-hovered{background-color:#313447;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus,html.theme--catppuccin-macchiato .content kbd.button:focus,html.theme--catppuccin-macchiato .button.is-dark.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark:focus:not(:active),html.theme--catppuccin-macchiato .content kbd.button:focus:not(:active),html.theme--catppuccin-macchiato .button.is-dark.is-focused:not(:active),html.theme--catppuccin-macchiato .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(54,58,79,0.25)}html.theme--catppuccin-macchiato .button.is-dark:active,html.theme--catppuccin-macchiato .content kbd.button:active,html.theme--catppuccin-macchiato .button.is-dark.is-active,html.theme--catppuccin-macchiato .content kbd.button.is-active{background-color:#2c2f40;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-dark[disabled],html.theme--catppuccin-macchiato .content kbd.button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button{background-color:#363a4f;border-color:#363a4f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-dark.is-inverted,html.theme--catppuccin-macchiato .content kbd.button.is-inverted{background-color:#fff;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted:hover,html.theme--catppuccin-macchiato 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.button.is-dark.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-focused,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-focused{background-color:#363a4f;border-color:#363a4f;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading::after{border-color:transparent transparent #363a4f #363a4f !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-dark.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-macchiato .content kbd.button.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-dark.is-outlined[disabled],html.theme--catppuccin-macchiato .content kbd.button.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-dark.is-outlined,fieldset[disabled] html.theme--catppuccin-macchiato .content kbd.button.is-outlined{background-color:transparent;border-color:#363a4f;box-shadow:none;color:#363a4f}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-macchiato .content 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.button.is-info.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-focused{background-color:#8bd5ca;border-color:#8bd5ca;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-outlined{background-color:transparent;border-color:#8bd5ca;box-shadow:none;color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#8bd5ca}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #8bd5ca #8bd5ca !important}html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-info.is-light{background-color:#f0faf8;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:hover,html.theme--catppuccin-macchiato .button.is-info.is-light.is-hovered{background-color:#e7f6f4;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-info.is-light:active,html.theme--catppuccin-macchiato .button.is-info.is-light.is-active{background-color:#ddf3f0;border-color:transparent;color:#276d62}html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:hover,html.theme--catppuccin-macchiato .button.is-success.is-hovered{background-color:#9ed78c;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus,html.theme--catppuccin-macchiato .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success:focus:not(:active),html.theme--catppuccin-macchiato .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,218,149,0.25)}html.theme--catppuccin-macchiato .button.is-success:active,html.theme--catppuccin-macchiato .button.is-success.is-active{background-color:#96d382;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success{background-color:#a6da95;border-color:#a6da95;box-shadow:none}html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-focused{background-color:#a6da95;border-color:#a6da95;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-outlined{background-color:transparent;border-color:#a6da95;box-shadow:none;color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6da95}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6da95 #a6da95 !important}html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-success.is-light{background-color:#f2faf0;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:hover,html.theme--catppuccin-macchiato .button.is-success.is-light.is-hovered{background-color:#eaf6e6;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-success.is-light:active,html.theme--catppuccin-macchiato .button.is-success.is-light.is-active{background-color:#e2f3dd;border-color:transparent;color:#386e26}html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:hover,html.theme--catppuccin-macchiato .button.is-warning.is-hovered{background-color:#eccf94;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus,html.theme--catppuccin-macchiato .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning:focus:not(:active),html.theme--catppuccin-macchiato .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(238,212,159,0.25)}html.theme--catppuccin-macchiato .button.is-warning:active,html.theme--catppuccin-macchiato .button.is-warning.is-active{background-color:#eaca89;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning{background-color:#eed49f;border-color:#eed49f;box-shadow:none}html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-focused{background-color:#eed49f;border-color:#eed49f;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-macchiato .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-outlined{background-color:transparent;border-color:#eed49f;box-shadow:none;color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#eed49f}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #eed49f #eed49f !important}html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-macchiato .button.is-warning.is-light{background-color:#fcf7ee;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:hover,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-hovered{background-color:#faf2e3;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-warning.is-light:active,html.theme--catppuccin-macchiato .button.is-warning.is-light.is-active{background-color:#f8eed8;border-color:transparent;color:#7e5c16}html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:hover,html.theme--catppuccin-macchiato .button.is-danger.is-hovered{background-color:#eb7c8c;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus,html.theme--catppuccin-macchiato .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger:focus:not(:active),html.theme--catppuccin-macchiato .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(237,135,150,0.25)}html.theme--catppuccin-macchiato .button.is-danger:active,html.theme--catppuccin-macchiato .button.is-danger.is-active{background-color:#ea7183;border-color:transparent;color:#fff}html.theme--catppuccin-macchiato .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger{background-color:#ed8796;border-color:#ed8796;box-shadow:none}html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-macchiato .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-focused{background-color:#ed8796;border-color:#ed8796;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-macchiato .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-outlined{background-color:transparent;border-color:#ed8796;box-shadow:none;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#ed8796}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #ed8796 #ed8796 !important}html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-macchiato .button.is-danger.is-light{background-color:#fcedef;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:hover,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-hovered{background-color:#fbe2e6;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-danger.is-light:active,html.theme--catppuccin-macchiato .button.is-danger.is-light.is-active{background-color:#f9d7dc;border-color:transparent;color:#971729}html.theme--catppuccin-macchiato .button.is-small,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-macchiato .button.is-small:not(.is-rounded),html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .button.is-normal{font-size:1rem}html.theme--catppuccin-macchiato .button.is-medium{font-size:1.25rem}html.theme--catppuccin-macchiato .button.is-large{font-size:1.5rem}html.theme--catppuccin-macchiato .button[disabled],fieldset[disabled] html.theme--catppuccin-macchiato .button{background-color:#6e738d;border-color:#5b6078;box-shadow:none;opacity:.5}html.theme--catppuccin-macchiato .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-macchiato .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-macchiato .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-macchiato .button.is-static{background-color:#1e2030;border-color:#5b6078;color:#8087a2;box-shadow:none;pointer-events:none}html.theme--catppuccin-macchiato .button.is-rounded,html.theme--catppuccin-macchiato #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-macchiato .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-macchiato .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-macchiato .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-macchiato .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-macchiato .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-macchiato .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-macchiato .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-macchiato .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-macchiato .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-macchiato .buttons.has-addons .button:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused,html.theme--catppuccin-macchiato .buttons.has-addons .button:active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-macchiato .buttons.has-addons .button:focus:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-focused:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button:active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-active:hover,html.theme--catppuccin-macchiato .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--catppuccin-macchiato .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--catppuccin-macchiato .buttons.is-centered{justify-content:center}html.theme--catppuccin-macchiato .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--catppuccin-macchiato 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kbd.button.is-hovered{background-color:#2c2d3d;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus,html.theme--catppuccin-mocha .content kbd.button:focus,html.theme--catppuccin-mocha .button.is-dark.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-dark:focus:not(:active),html.theme--catppuccin-mocha .content kbd.button:focus:not(:active),html.theme--catppuccin-mocha .button.is-dark.is-focused:not(:active),html.theme--catppuccin-mocha .content kbd.button.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(49,50,68,0.25)}html.theme--catppuccin-mocha .button.is-dark:active,html.theme--catppuccin-mocha .content kbd.button:active,html.theme--catppuccin-mocha .button.is-dark.is-active,html.theme--catppuccin-mocha .content kbd.button.is-active{background-color:#262735;border-color:transparent;color:#fff}html.theme--catppuccin-mocha 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kbd.button.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-focused,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-focused{background-color:#fff;color:#313244}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined.is-loading.is-focused::after,html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #313244 #313244 !important}html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined[disabled],html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-dark.is-inverted.is-outlined,fieldset[disabled] html.theme--catppuccin-mocha .content kbd.button.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-primary,html.theme--catppuccin-mocha .docstring>section>a.button.docs-sourcelink{background-color:#89b4fa;border-color:transparent;color:#fff}html.theme--catppuccin-mocha 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html.theme--catppuccin-mocha .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-link.is-light{background-color:#ebf3fe;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:hover,html.theme--catppuccin-mocha .button.is-link.is-light.is-hovered{background-color:#dfebfe;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-link.is-light:active,html.theme--catppuccin-mocha .button.is-link.is-light.is-active{background-color:#d3e3fd;border-color:transparent;color:#063c93}html.theme--catppuccin-mocha .button.is-info{background-color:#94e2d5;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:hover,html.theme--catppuccin-mocha .button.is-info.is-hovered{background-color:#8adfd1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info:focus,html.theme--catppuccin-mocha 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.button.is-info.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-focused{background-color:#94e2d5;border-color:#94e2d5;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-info.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-outlined{background-color:transparent;border-color:#94e2d5;box-shadow:none;color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#94e2d5}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #94e2d5 #94e2d5 !important}html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-info.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-info.is-light{background-color:#effbf9;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:hover,html.theme--catppuccin-mocha .button.is-info.is-light.is-hovered{background-color:#e5f8f5;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-info.is-light:active,html.theme--catppuccin-mocha .button.is-info.is-light.is-active{background-color:#dbf5f1;border-color:transparent;color:#207466}html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:hover,html.theme--catppuccin-mocha .button.is-success.is-hovered{background-color:#9de097;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus,html.theme--catppuccin-mocha .button.is-success.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success:focus:not(:active),html.theme--catppuccin-mocha .button.is-success.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(166,227,161,0.25)}html.theme--catppuccin-mocha .button.is-success:active,html.theme--catppuccin-mocha .button.is-success.is-active{background-color:#93dd8d;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success{background-color:#a6e3a1;border-color:#a6e3a1;box-shadow:none}html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-focused{background-color:#a6e3a1;border-color:#a6e3a1;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-outlined{background-color:transparent;border-color:#a6e3a1;box-shadow:none;color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#a6e3a1}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #a6e3a1 #a6e3a1 !important}html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-success.is-light{background-color:#f0faef;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:hover,html.theme--catppuccin-mocha .button.is-success.is-light.is-hovered{background-color:#e7f7e5;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-success.is-light:active,html.theme--catppuccin-mocha .button.is-success.is-light.is-active{background-color:#def4dc;border-color:transparent;color:#287222}html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:hover,html.theme--catppuccin-mocha .button.is-warning.is-hovered{background-color:#f8dea3;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus,html.theme--catppuccin-mocha .button.is-warning.is-focused{border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning:focus:not(:active),html.theme--catppuccin-mocha .button.is-warning.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(249,226,175,0.25)}html.theme--catppuccin-mocha .button.is-warning:active,html.theme--catppuccin-mocha .button.is-warning.is-active{background-color:#f7d997;border-color:transparent;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning{background-color:#f9e2af;border-color:#f9e2af;box-shadow:none}html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-hovered{background-color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted{background-color:rgba(0,0,0,0.7);border-color:transparent;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-loading::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-focused{background-color:#f9e2af;border-color:#f9e2af;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-outlined.is-loading.is-focused::after{border-color:transparent transparent rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--catppuccin-mocha .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-outlined{background-color:transparent;border-color:#f9e2af;box-shadow:none;color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f9e2af}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f9e2af #f9e2af !important}html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--catppuccin-mocha .button.is-warning.is-light{background-color:#fef8ec;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:hover,html.theme--catppuccin-mocha .button.is-warning.is-light.is-hovered{background-color:#fdf4e0;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-warning.is-light:active,html.theme--catppuccin-mocha .button.is-warning.is-light.is-active{background-color:#fcf0d4;border-color:transparent;color:#8a620a}html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:hover,html.theme--catppuccin-mocha .button.is-danger.is-hovered{background-color:#f27f9f;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus,html.theme--catppuccin-mocha .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger:focus:not(:active),html.theme--catppuccin-mocha .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(243,139,168,0.25)}html.theme--catppuccin-mocha .button.is-danger:active,html.theme--catppuccin-mocha .button.is-danger.is-active{background-color:#f17497;border-color:transparent;color:#fff}html.theme--catppuccin-mocha .button.is-danger[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger{background-color:#f38ba8;border-color:#f38ba8;box-shadow:none}html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--catppuccin-mocha .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-focused{background-color:#f38ba8;border-color:#f38ba8;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--catppuccin-mocha .button.is-danger.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-outlined{background-color:transparent;border-color:#f38ba8;box-shadow:none;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:hover,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-hovered,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined:focus,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-focused{background-color:#fff;color:#f38ba8}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:hover::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading:focus::after,html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f38ba8 #f38ba8 !important}html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--catppuccin-mocha .button.is-danger.is-light{background-color:#fdedf1;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:hover,html.theme--catppuccin-mocha .button.is-danger.is-light.is-hovered{background-color:#fce1e8;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-danger.is-light:active,html.theme--catppuccin-mocha .button.is-danger.is-light.is-active{background-color:#fbd5e0;border-color:transparent;color:#991036}html.theme--catppuccin-mocha .button.is-small,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--catppuccin-mocha .button.is-small:not(.is-rounded),html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .button.is-normal{font-size:1rem}html.theme--catppuccin-mocha .button.is-medium{font-size:1.25rem}html.theme--catppuccin-mocha .button.is-large{font-size:1.5rem}html.theme--catppuccin-mocha .button[disabled],fieldset[disabled] html.theme--catppuccin-mocha .button{background-color:#6c7086;border-color:#585b70;box-shadow:none;opacity:.5}html.theme--catppuccin-mocha .button.is-fullwidth{display:flex;width:100%}html.theme--catppuccin-mocha .button.is-loading{color:transparent !important;pointer-events:none}html.theme--catppuccin-mocha .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--catppuccin-mocha .button.is-static{background-color:#181825;border-color:#585b70;color:#7f849c;box-shadow:none;pointer-events:none}html.theme--catppuccin-mocha .button.is-rounded,html.theme--catppuccin-mocha #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 0.25em);padding-right:calc(1em + 0.25em)}html.theme--catppuccin-mocha .buttons{align-items:center;display:flex;flex-wrap:wrap;justify-content:flex-start}html.theme--catppuccin-mocha .buttons .button{margin-bottom:0.5rem}html.theme--catppuccin-mocha .buttons .button:not(:last-child):not(.is-fullwidth){margin-right:.5rem}html.theme--catppuccin-mocha .buttons:last-child{margin-bottom:-0.5rem}html.theme--catppuccin-mocha .buttons:not(:last-child){margin-bottom:1rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large){font-size:.75rem}html.theme--catppuccin-mocha .buttons.are-small .button:not(.is-normal):not(.is-medium):not(.is-large):not(.is-rounded){border-radius:3px}html.theme--catppuccin-mocha .buttons.are-medium .button:not(.is-small):not(.is-normal):not(.is-large){font-size:1.25rem}html.theme--catppuccin-mocha .buttons.are-large .button:not(.is-small):not(.is-normal):not(.is-medium){font-size:1.5rem}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--catppuccin-mocha .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--catppuccin-mocha .buttons.has-addons .button:last-child{margin-right:0}html.theme--catppuccin-mocha .buttons.has-addons .button:hover,html.theme--catppuccin-mocha .buttons.has-addons .button.is-hovered{z-index:2}html.theme--catppuccin-mocha .buttons.has-addons .button:focus,html.theme--catppuccin-mocha .buttons.has-addons .button.is-focused,html.theme--catppuccin-mocha .buttons.has-addons .button:active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-active,html.theme--catppuccin-mocha .buttons.has-addons .button.is-selected{z-index:3}html.theme--catppuccin-mocha .buttons.has-addons 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.button.is-link.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-link.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-outlined{background-color:transparent;border-color:#1abc9c;box-shadow:none;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-focused{background-color:#fff;color:#1abc9c}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-link.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #1abc9c #1abc9c !important}html.theme--documenter-dark .button.is-link.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-link.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-link.is-light{background-color:#edfdf9;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:hover,html.theme--documenter-dark .button.is-link.is-light.is-hovered{background-color:#e2fbf6;border-color:transparent;color:#15987e}html.theme--documenter-dark .button.is-link.is-light:active,html.theme--documenter-dark 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.button.is-info.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-focused{background-color:#fff;color:#3c5dcd}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-info.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #3c5dcd #3c5dcd !important}html.theme--documenter-dark .button.is-info.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark 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.button.is-success.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;color:#259a12}html.theme--documenter-dark .button.is-success.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-outlined.is-focused{background-color:#259a12;border-color:#259a12;color:#fff}html.theme--documenter-dark .button.is-success.is-outlined.is-loading::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-success.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-outlined{background-color:transparent;border-color:#259a12;box-shadow:none;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;color:#fff}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-focused{background-color:#fff;color:#259a12}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-success.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #259a12 #259a12 !important}html.theme--documenter-dark .button.is-success.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-success.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-success.is-light{background-color:#effded;color:#2ec016}html.theme--documenter-dark .button.is-success.is-light:hover,html.theme--documenter-dark 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rgba(0,0,0,0.7) rgba(0,0,0,0.7) !important}html.theme--documenter-dark .button.is-warning.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-outlined{background-color:transparent;border-color:#f4c72f;box-shadow:none;color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:hover,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-hovered,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined:focus,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-focused{background-color:rgba(0,0,0,0.7);color:#f4c72f}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:hover::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-hovered::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading:focus::after,html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined.is-loading.is-focused::after{border-color:transparent transparent #f4c72f #f4c72f !important}html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-warning.is-inverted.is-outlined{background-color:transparent;border-color:rgba(0,0,0,0.7);box-shadow:none;color:rgba(0,0,0,0.7)}html.theme--documenter-dark .button.is-warning.is-light{background-color:#fefaec;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:hover,html.theme--documenter-dark .button.is-warning.is-light.is-hovered{background-color:#fdf7e0;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-warning.is-light:active,html.theme--documenter-dark .button.is-warning.is-light.is-active{background-color:#fdf3d3;border-color:transparent;color:#8c6e07}html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:hover,html.theme--documenter-dark .button.is-danger.is-hovered{background-color:#c13930;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus,html.theme--documenter-dark .button.is-danger.is-focused{border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger:focus:not(:active),html.theme--documenter-dark .button.is-danger.is-focused:not(:active){box-shadow:0 0 0 0.125em rgba(203,60,51,0.25)}html.theme--documenter-dark .button.is-danger:active,html.theme--documenter-dark .button.is-danger.is-active{background-color:#b7362e;border-color:transparent;color:#fff}html.theme--documenter-dark .button.is-danger[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger{background-color:#cb3c33;border-color:#cb3c33;box-shadow:none}html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-inverted:hover,html.theme--documenter-dark .button.is-danger.is-inverted.is-hovered{background-color:#f2f2f2}html.theme--documenter-dark .button.is-danger.is-inverted[disabled],fieldset[disabled] html.theme--documenter-dark .button.is-danger.is-inverted{background-color:#fff;border-color:transparent;box-shadow:none;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-loading::after{border-color:transparent transparent #fff #fff !important}html.theme--documenter-dark .button.is-danger.is-outlined{background-color:transparent;border-color:#cb3c33;color:#cb3c33}html.theme--documenter-dark .button.is-danger.is-outlined:hover,html.theme--documenter-dark .button.is-danger.is-outlined.is-hovered,html.theme--documenter-dark 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html.theme--documenter-dark .button.is-danger.is-inverted.is-outlined{background-color:transparent;border-color:#fff;box-shadow:none;color:#fff}html.theme--documenter-dark .button.is-danger.is-light{background-color:#fbefef;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:hover,html.theme--documenter-dark .button.is-danger.is-light.is-hovered{background-color:#f8e6e5;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-danger.is-light:active,html.theme--documenter-dark .button.is-danger.is-light.is-active{background-color:#f6dcda;border-color:transparent;color:#c03930}html.theme--documenter-dark .button.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{font-size:.75rem}html.theme--documenter-dark .button.is-small:not(.is-rounded),html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button:not(.is-rounded){border-radius:3px}html.theme--documenter-dark .button.is-normal{font-size:1rem}html.theme--documenter-dark .button.is-medium{font-size:1.25rem}html.theme--documenter-dark .button.is-large{font-size:1.5rem}html.theme--documenter-dark .button[disabled],fieldset[disabled] html.theme--documenter-dark .button{background-color:#8c9b9d;border-color:#5e6d6f;box-shadow:none;opacity:.5}html.theme--documenter-dark .button.is-fullwidth{display:flex;width:100%}html.theme--documenter-dark .button.is-loading{color:transparent !important;pointer-events:none}html.theme--documenter-dark .button.is-loading::after{position:absolute;left:calc(50% - (1em * 0.5));top:calc(50% - (1em * 0.5));position:absolute !important}html.theme--documenter-dark .button.is-static{background-color:#282f2f;border-color:#5e6d6f;color:#dbdee0;box-shadow:none;pointer-events:none}html.theme--documenter-dark .button.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.button{border-radius:9999px;padding-left:calc(1em + 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.buttons.has-addons .button:not(:first-child){border-bottom-left-radius:0;border-top-left-radius:0}html.theme--documenter-dark .buttons.has-addons .button:not(:last-child){border-bottom-right-radius:0;border-top-right-radius:0;margin-right:-1px}html.theme--documenter-dark .buttons.has-addons .button:last-child{margin-right:0}html.theme--documenter-dark .buttons.has-addons .button:hover,html.theme--documenter-dark .buttons.has-addons .button.is-hovered{z-index:2}html.theme--documenter-dark .buttons.has-addons .button:focus,html.theme--documenter-dark .buttons.has-addons .button.is-focused,html.theme--documenter-dark .buttons.has-addons .button:active,html.theme--documenter-dark .buttons.has-addons .button.is-active,html.theme--documenter-dark .buttons.has-addons .button.is-selected{z-index:3}html.theme--documenter-dark .buttons.has-addons .button:focus:hover,html.theme--documenter-dark .buttons.has-addons .button.is-focused:hover,html.theme--documenter-dark .buttons.has-addons .button:active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-active:hover,html.theme--documenter-dark .buttons.has-addons .button.is-selected:hover{z-index:4}html.theme--documenter-dark .buttons.has-addons .button.is-expanded{flex-grow:1;flex-shrink:1}html.theme--documenter-dark .buttons.is-centered{justify-content:center}html.theme--documenter-dark .buttons.is-centered:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}html.theme--documenter-dark .buttons.is-right{justify-content:flex-end}html.theme--documenter-dark .buttons.is-right:not(.has-addons) .button:not(.is-fullwidth){margin-left:0.25rem;margin-right:0.25rem}@media screen and (max-width: 768px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.5625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1rem}}@media screen and (min-width: 769px) and (max-width: 1055px){html.theme--documenter-dark .button.is-responsive.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.is-responsive{font-size:.65625rem}html.theme--documenter-dark .button.is-responsive,html.theme--documenter-dark .button.is-responsive.is-normal{font-size:.75rem}html.theme--documenter-dark .button.is-responsive.is-medium{font-size:1rem}html.theme--documenter-dark .button.is-responsive.is-large{font-size:1.25rem}}html.theme--documenter-dark .container{flex-grow:1;margin:0 auto;position:relative;width:auto}html.theme--documenter-dark .container.is-fluid{max-width:none !important;padding-left:32px;padding-right:32px;width:100%}@media screen and (min-width: 1056px){html.theme--documenter-dark .container{max-width:992px}}@media screen and (max-width: 1215px){html.theme--documenter-dark .container.is-widescreen:not(.is-max-desktop){max-width:1152px}}@media screen and (max-width: 1407px){html.theme--documenter-dark .container.is-fullhd:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}@media screen and (min-width: 1216px){html.theme--documenter-dark .container:not(.is-max-desktop){max-width:1152px}}@media screen and (min-width: 1408px){html.theme--documenter-dark .container:not(.is-max-desktop):not(.is-max-widescreen){max-width:1344px}}html.theme--documenter-dark .content li+li{margin-top:0.25em}html.theme--documenter-dark .content p:not(:last-child),html.theme--documenter-dark .content dl:not(:last-child),html.theme--documenter-dark .content ol:not(:last-child),html.theme--documenter-dark .content ul:not(:last-child),html.theme--documenter-dark .content blockquote:not(:last-child),html.theme--documenter-dark .content pre:not(:last-child),html.theme--documenter-dark .content table:not(:last-child){margin-bottom:1em}html.theme--documenter-dark .content h1,html.theme--documenter-dark .content h2,html.theme--documenter-dark .content h3,html.theme--documenter-dark .content h4,html.theme--documenter-dark .content h5,html.theme--documenter-dark .content h6{color:#f2f2f2;font-weight:600;line-height:1.125}html.theme--documenter-dark .content h1{font-size:2em;margin-bottom:0.5em}html.theme--documenter-dark .content h1:not(:first-child){margin-top:1em}html.theme--documenter-dark .content h2{font-size:1.75em;margin-bottom:0.5714em}html.theme--documenter-dark .content h2:not(:first-child){margin-top:1.1428em}html.theme--documenter-dark .content h3{font-size:1.5em;margin-bottom:0.6666em}html.theme--documenter-dark .content h3:not(:first-child){margin-top:1.3333em}html.theme--documenter-dark .content h4{font-size:1.25em;margin-bottom:0.8em}html.theme--documenter-dark .content h5{font-size:1.125em;margin-bottom:0.8888em}html.theme--documenter-dark .content h6{font-size:1em;margin-bottom:1em}html.theme--documenter-dark .content blockquote{background-color:#282f2f;border-left:5px solid #5e6d6f;padding:1.25em 1.5em}html.theme--documenter-dark .content ol{list-style-position:outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ol:not([type]){list-style-type:decimal}html.theme--documenter-dark .content ol.is-lower-alpha:not([type]){list-style-type:lower-alpha}html.theme--documenter-dark .content ol.is-lower-roman:not([type]){list-style-type:lower-roman}html.theme--documenter-dark .content ol.is-upper-alpha:not([type]){list-style-type:upper-alpha}html.theme--documenter-dark .content ol.is-upper-roman:not([type]){list-style-type:upper-roman}html.theme--documenter-dark .content ul{list-style:disc outside;margin-left:2em;margin-top:1em}html.theme--documenter-dark .content ul ul{list-style-type:circle;margin-top:0.5em}html.theme--documenter-dark .content ul ul ul{list-style-type:square}html.theme--documenter-dark .content dd{margin-left:2em}html.theme--documenter-dark .content figure{margin-left:2em;margin-right:2em;text-align:center}html.theme--documenter-dark .content figure:not(:first-child){margin-top:2em}html.theme--documenter-dark .content figure:not(:last-child){margin-bottom:2em}html.theme--documenter-dark .content figure img{display:inline-block}html.theme--documenter-dark .content figure figcaption{font-style:italic}html.theme--documenter-dark .content pre{-webkit-overflow-scrolling:touch;overflow-x:auto;padding:0;white-space:pre;word-wrap:normal}html.theme--documenter-dark .content sup,html.theme--documenter-dark .content sub{font-size:75%}html.theme--documenter-dark .content table{width:100%}html.theme--documenter-dark .content table td,html.theme--documenter-dark .content table th{border:1px solid #5e6d6f;border-width:0 0 1px;padding:0.5em 0.75em;vertical-align:top}html.theme--documenter-dark .content table th{color:#f2f2f2}html.theme--documenter-dark .content table th:not([align]){text-align:inherit}html.theme--documenter-dark .content table thead td,html.theme--documenter-dark .content table thead th{border-width:0 0 2px;color:#f2f2f2}html.theme--documenter-dark .content table tfoot td,html.theme--documenter-dark .content table tfoot th{border-width:2px 0 0;color:#f2f2f2}html.theme--documenter-dark .content table tbody tr:last-child td,html.theme--documenter-dark .content table tbody tr:last-child th{border-bottom-width:0}html.theme--documenter-dark .content .tabs li+li{margin-top:0}html.theme--documenter-dark .content.is-small,html.theme--documenter-dark #documenter .docs-sidebar form.docs-search>input.content{font-size:.75rem}html.theme--documenter-dark .content.is-normal{font-size:1rem}html.theme--documenter-dark .content.is-medium{font-size:1.25rem}html.theme--documenter-dark .content.is-large{font-size:1.5rem}html.theme--documenter-dark 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img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img{display:block;height:auto;width:100%}html.theme--documenter-dark .image img.is-rounded,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img img.is-rounded{border-radius:9999px}html.theme--documenter-dark .image.is-fullwidth,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-fullwidth{width:100%}html.theme--documenter-dark .image.is-square img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square img,html.theme--documenter-dark .image.is-square .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-square .has-ratio,html.theme--documenter-dark .image.is-1by1 img,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 img,html.theme--documenter-dark .image.is-1by1 .has-ratio,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by1 .has-ratio,html.theme--documenter-dark .image.is-5by4 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.docs-logo>img.is-3by1{padding-top:33.3333%}html.theme--documenter-dark .image.is-4by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-4by5{padding-top:125%}html.theme--documenter-dark .image.is-3by4,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by4{padding-top:133.3333%}html.theme--documenter-dark .image.is-2by3,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-2by3{padding-top:150%}html.theme--documenter-dark .image.is-3by5,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-3by5{padding-top:166.6666%}html.theme--documenter-dark .image.is-9by16,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-9by16{padding-top:177.7777%}html.theme--documenter-dark .image.is-1by2,html.theme--documenter-dark #documenter .docs-sidebar .docs-logo>img.is-1by2{padding-top:200%}html.theme--documenter-dark .image.is-1by3,html.theme--documenter-dark #documenter .docs-sidebar 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.hljs-string{color:#38a}.hljs-emphasis{font-style:italic}.hljs-strong{font-weight:bold}.gap-4{gap:1rem} diff --git a/previews/PR184/assets/themeswap.js b/previews/PR184/assets/themeswap.js new file mode 100644 index 00000000..9f5eebe6 --- /dev/null +++ b/previews/PR184/assets/themeswap.js @@ -0,0 +1,84 @@ +// Small function to quickly swap out themes. Gets put into the tag.. +function set_theme_from_local_storage() { + // Initialize the theme to null, which means default + var theme = null; + // If the browser supports the localstorage and is not disabled then try to get the + // documenter theme + if (window.localStorage != null) { + // Get the user-picked theme from localStorage. May be `null`, which means the default + // theme. + theme = window.localStorage.getItem("documenter-theme"); + } + // Check if the users preference is for dark color scheme + var darkPreference = + window.matchMedia("(prefers-color-scheme: dark)").matches === true; + // Initialize a few variables for the loop: + // + // - active: will contain the index of the theme that should be active. Note that there + // is no guarantee that localStorage contains sane values. If `active` stays `null` + // we either could not find the theme or it is the default (primary) theme anyway. + // Either way, we then need to stick to the primary theme. + // + // - disabled: style sheets that should be disabled (i.e. all the theme style sheets + // that are not the currently active theme) + var active = null; + var disabled = []; + var primaryLightTheme = null; + var primaryDarkTheme = null; + for (var i = 0; i < document.styleSheets.length; i++) { + var ss = document.styleSheets[i]; + // The tag of each style sheet is expected to have a data-theme-name attribute + // which must contain the name of the theme. The names in localStorage much match this. + var themename = ss.ownerNode.getAttribute("data-theme-name"); + // attribute not set => non-theme stylesheet => ignore + if (themename === null) continue; + // To distinguish the default (primary) theme, it needs to have the data-theme-primary + // attribute set. + if (ss.ownerNode.getAttribute("data-theme-primary") !== null) { + primaryLightTheme = themename; + } + // Check if the theme is primary dark theme so that we could store its name in darkTheme + if (ss.ownerNode.getAttribute("data-theme-primary-dark") !== null) { + primaryDarkTheme = themename; + } + // If we find a matching theme (and it's not the default), we'll set active to non-null + if (themename === theme) active = i; + // Store the style sheets of inactive themes so that we could disable them + if (themename !== theme) disabled.push(ss); + } + var activeTheme = null; + if (active !== null) { + // If we did find an active theme, we'll (1) add the theme--$(theme) class to + document.getElementsByTagName("html")[0].className = "theme--" + theme; + activeTheme = theme; + } else { + // If we did _not_ find an active theme, then we need to fall back to the primary theme + // which can either be dark or light, depending on the user's OS preference. + var activeTheme = darkPreference ? primaryDarkTheme : primaryLightTheme; + // In case it somehow happens that the relevant primary theme was not found in the + // preceding loop, we abort without doing anything. + if (activeTheme === null) { + console.error("Unable to determine primary theme."); + return; + } + // When switching to the primary light theme, then we must not have a class name + // for the tag. 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Click here to go to the documentation for the latest stable release.'; + div.appendChild(closer); + document.body.appendChild(div); +} + +if (document.readyState === "loading") { + document.addEventListener("DOMContentLoaded", maybeAddWarning); +} else { + maybeAddWarning(); +} diff --git a/previews/PR184/generated/StochasticSystem.jl b/previews/PR184/generated/StochasticSystem.jl new file mode 100644 index 00000000..4edbe34b --- /dev/null +++ b/previews/PR184/generated/StochasticSystem.jl @@ -0,0 +1,128 @@ +# # SDE Tutorial +# +# +# #### Topics covered in this tutorial include: +# +# - the swing equation +# - fixpoint search of ODE systems +# - constructing an SDE problem with NetworkDynamics.jl +# +# ## Example: Fluctuations in Power Grids +# +# This tutorial explains the use of stochastic differential equations (SDE's) in NetworkDynamics.jl. As an example system we will simulate fluctuations in a simple power grid model. +# +# The phase and frequency dynamics in power grids can be modeled by the swing equation. In the complex systems community this is also known as the Kuramoto model with inertia. +# +# ```math +# \begin{aligned} +# \dot \theta_i &= \omega_i \\ +# M_i \dot \omega_i &= P_i(t) - D_i \omega_i - \sum_{i=1}^N K_{ij} \sin(\theta_i - \theta_j) +# \end{aligned} +# ``` +# +# Here, $M_i$ and $D_i$ are inertia and damping parameters, $K_{ij}$ is the power capacitiy of the line and $P_i(t)$ is the power infeed at the node. We assume that this power can be separated into a constant power setpoint $P^\circ_i$ and a stochastic fluctuation. +# +# ```math +# \begin{aligned} +# P_i(t) = P^\circ_i + \sigma_i \cdot \xi_i(t) +# \end{aligned} +# ``` +# +# In this tutorial we assume the fluctuations $\xi_i(t)$ to be white Gaussian noise. Separating the deterministic and stochastic part, we can write this problem as a stochastic differential equation. +# +# ```math +# \begin{aligned} +# d\omega_i = \left[P_i - D_i \omega_i - \sum_{i=1}^N K_{ij} \sin(\theta_i - \theta_j)\right] dt + \sigma_i dW_i +# \end{aligned} +# ``` +# +# Here, $dW_i = \xi_i dt$ is the infinitesimal increment of the Wiener process. Problems of this type can be numerically solved in Julia as described in the [SDE Tutorial](https://diffeq.sciml.ai/stable/tutorials/sde_example/) of `DifferentialEquations`. +# +# ## Implementing the Swing Equation +# +# First we will implement the node and edge models for the deterministic case +# without the fluctuations. We set the defaults for the inertia and damping +# parameters to be $M_i = 1.0$ and $D_i = 0.1$. The default coupling strength is $K=6$. + +using NetworkDynamics, Graphs +using StochasticDiffEq, OrdinaryDiffEqTsit5 +using Plots, LaTeXStrings + +function swing_equation!(dv, v, esum, (M, P, D), t) + dv[1] = v[2] + dv[2] = 1/M *(P - D * v[2] + esum[1]) + nothing +end +swing_vertex = VertexModel(f=swing_equation!, g=1, sym=[:θ, :ω], psym=[:M=>1, :P, :D=>0.1]) + +function powerflow!(e, v_s, v_d, (K,), t) + e[1] = K * sin(v_s[1] - v_d[1]) +end +powerflow_edge = EdgeModel(g=AntiSymmetric(powerflow!), outdim=1, psym=[:K=>6]) + +# ## Contructing the Deterministic Dynamics +# +# For the graph structure we will use a simple 4 node ring network. + +g = watts_strogatz(4, 2, 0.0) + +# Then we can construct the `Network` of the deterministic system. + +nd = Network(g, swing_vertex, powerflow_edge) + +# ## Fixpoint Search +# +# Now we need to define the dynamic parameters of vertices and edges. +# For that, we start by creating the `NWParameter` object, pre filling it with the default values: + +p = NWParameter(nd) + +# Most of the parameters are allready set +# For the nodes we assume half of them to be net producers ($P = 1.0$) and half of them to be net consumers ($P = -1.0$) of power. + +p.v[1:4, :P] = [1, -1, 1, -1] + +# We want to simulate fluctuations around an equilibrium state of our model +# system. Therefore, we need to find a fixpoint of the determinitic system which +# can be done by using the utility function `find_fixpoint()`. As an initial guess +# we take all variables equal to zero. + +u0 = find_fixpoint(nd, p) + +ode_prob = ODEProblem(nd, uflat(u0), (0.0, 500.0), pflat(p)) +ode_sol = solve(ode_prob, Tsit5()) + +plot(ode_sol; idxs=vidxs(nd,:,:ω), ylims=(-1.0, 1.0), ylabel=L"\omega", legend=false, fmt=:png) + +# We see that this is in fact a fixpoint solution. We will later use this as an initial condition for the numerical integration of the SDE system. +# +# ## Adding a Stochastic Layer +# +# For adding the stochastic part of the dynamics we have to define a second graph layer. In our example, the fluctuations at different nodes are independent of each other. Therefore, we define a second graph with the same number of vertices but without any edges. + +h = SimpleGraph(4, 0) + +# The dynamics at the nodes has to have the same dimension as in the deterministic case. In our example we only have fluctuations in the second variable. +# FIXME: stochastic system should be simulated using symbolic indexing, this is jsut wrong in current ND! + +# function fluctuation!(dx, x, edges, p, t) +# dx[1] = 0.0 +# dx[2] = 0.05 +# end + +# Now we can construct the dynamics of the second layer by using `network_dynamics()`. Since the graph structure of the stochastic layer has no edges we can take the edge model of the deterministic case as a placeholder. + +# fluctuation_vertex = VertexModel(f=fluctuation!, g=1:2, dim=2) +# nd_noise = Network(h, fluctuation_vertex, NetworkDynamics.EdgeModel[]) + +# ## Simulating the SDE +# +# Finally, we can create an `SDEProblem` and solve it with `DifferentialEquations`. + +# sde_prob = SDEProblem(nd, nd_noise, uflat(u0), (0.0, 500.0), pflat(p)) +# sde_sol = solve(sde_prob, SOSRA()) +# plot(sde_sol; idxs=vidxs(nd,:,:ω), ylims=(-1.0, 1.0), ylabel=L"\omega", legend=false, fmt=:png) + +# More details on SDE problems, e.g. how to include correlations or how to define an `EnsembleProblem`, can be found in the [documentation](https://diffeq.sciml.ai/stable/types/sde_types/) of `DifferentialEquations`. + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/cascading_failure.jl b/previews/PR184/generated/cascading_failure.jl new file mode 100644 index 00000000..2e3f3162 --- /dev/null +++ b/previews/PR184/generated/cascading_failure.jl @@ -0,0 +1,152 @@ +# # Cascading Failure +# This script reimplements the minimal example of a dynamic cascading failure +# described in Schäfer et al. (2018) [1]. +# In is an example how to use callback functions to change network parameters. In +# this case to disable certain lines. +# +# > [1] Schäfer, B., Witthaut, D., Timme, M., & Latora, V. (2018). Dynamically induced cascading failures in power grids. Nature communications, 9(1), 1-13. https://www.nature.com/articles/s41467-018-04287-5 +# +# The system is modeled using swing equation and active power edges. The nodes are +# characterized by the voltage angle `δ`, the active power on each line is symmetric +# and a function of the difference between source and destination angle `δ_src - δ_dst`. + +using NetworkDynamics +using Graphs +using OrdinaryDiffEqTsit5 +using DiffEqCallbacks +using Plots +using Test #hide +import SymbolicIndexingInterface as SII + +# For the nodes we define the swing equation. State `v[1] = δ`, `v[2] = ω`. +# The swing equation has three parameters: `p = (P_ref, I, γ)` where `P_ref` +# is the power setpopint, `I` is the inertia and `γ` is the droop or damping coeficcient. +# +# The output of the node is just the first state. `g=1` is a shorthand for `g=StateMask(1:1)` +# which implements a trivial output function `g` which just takes the first element of the +# state vector. + +function swing_equation(dv, v, esum, p,t) + P, I, γ = p + dv[1] = v[2] + dv[2] = P - γ * v[2] .+ esum[1] + dv[2] = dv[2] / I + nothing +end +vertex = VertexModel(f=swing_equation, g=1, sym=[:δ, :ω], psym=[:P_ref, :I=>1, :γ=>0.1]) + +# Lets define a simple purely active power line whose active power flow is +# completlye determined by the connected voltage angles and the coupling constant +# `K`. +# We give an additonal parameter, the line limit, which we'll use later in the callback. + +function simple_edge(e, v_s, v_d, (K,), t) + e[1] = K * sin(v_s[1] - v_d[1]) +end +edge = EdgeModel(;g=AntiSymmetric(simple_edge), outsym=:P, psym=[:K=>1.63, :limit=>1]) + +# With the definition of the graph topology we can build the `Network` object: + +g = SimpleGraph([0 1 1 0 1; + 1 0 1 1 0; + 1 1 0 1 0; + 0 1 1 0 1; + 1 0 0 1 0]) +swing_network = Network(g, vertex, edge) + +# For the parameters, we create the `NWParameter` object prefilled with default p values + +p = NWParameter(swing_network) +# vertices 1, 3 and 4 act as loads +p.v[(1,3,4), :P_ref] .= -1 +# vertices 2 and 5 act as generators +p.v[(2,5), :P_ref] .= 1.5 + +# We can use `find_fixpoint` to find a valid initial condition of the network + +u0 = find_fixpoint(swing_network, p) + +# In order to implement the line failures, we need to create a `VectorContinousCallback`. +# In the callback, we compare the current flow on the line with the limit. If the limit is reached, +# the coupling `K` is set to 0. +# +# First we can define the affect function: + +function affect!(integrator, idx) + println("Line $idx tripped at t=$(integrator.t)") + p = NWParameter(integrator) # get indexable parameter object + p.e[idx, :K] = 0 + auto_dt_reset!(integrator) + save_parameters!(integrator) + nothing +end + +# There is one important aspect to this function: the `save_parameters!` call. +# In the callback, we change the parameters of the network, making the parameters time +# dependent. The flow on the line is a function `P(t) = f(u(t), p(t))`. Thus we need to +# inform the integrator, that a discrete change in parameters happend. With this, the +# solution object not only tracks `u(t)` but also `p(t)` and we may extract the observable `P(t)` +# directly. +# +# The callback trigger condition is a bit more complicated. The straight forward version looks like this: + +function naive_condition(out, u, t, integrator) + # careful, u != integrator.u + # therefore construct nwstate with Network info from integrator but u + s = NWState(integrator, u, integrator.p, t) + for i in eachindex(out) + out[i] = abs(s.e[i,:P]) - s.p.e[1,:limit] # compare flow with limit for line + end + nothing +end + +# However, from a performacne perspectiv there are problems with this solution: on +# every call, we need to perform symbolic indexing into the `NWState` object. +# Symbolic indexing is not cheap, as it requires to gather meta data about the network. +# Luckily, the `SymbolicIndexingInterface` package which powers the symbolic indexing +# provides the lower level functions `getp` and `getu` which can be used to create and +# cache accessors to the internal states. +# +# This still isn't ideal beacuse both `getlim` and `getflow` getters will create arrays +# within the callback. But is far better then resolving the flat state indices every time. + +condition = let getlim = SII.getp(swing_network, epidxs(swing_network, :, :limit)), + getflow = SII.getu(swing_network, eidxs(swing_network, :, :P)) + function (out, u, t, integrator) + # careful, u != integrator.u + # therefore construct nwstate with Network info from integrator but u + s = NWState(integrator, u, integrator.p, t) + out .= getlim(s) .- abs.(getflow(s)) + nothing + end +end + +# We can combine affect and condition to form the callback. + +trip_cb = VectorContinuousCallback(condition, affect!, ne(g)); + +# However, there is another component missing. If we look at the powerflow on the +# lines in the initial steady state + +u0.e[:, :P] + +# We see that every flow is below the trip value 1.0. Therefor we need to add a distrubance +# to the network. We do this by manually disabeling line 5 at time 1. + +trip_first_cb = PresetTimeCallback(1.0, integrator->affect!(integrator, 5)); + +# With those components, we can create the problem and solve it. + +prob = ODEProblem(swing_network, uflat(u0), (0,6), copy(pflat(p)); + callback=CallbackSet(trip_cb, trip_first_cb)) +sol = solve(prob, Tsit5()); +# we want to test the reconstruction of the observables # hide +@test all(!iszero, sol(sol.t; idxs=eidxs(sol,:,:P))[begin]) # hide +@test all(iszero, sol(sol.t; idxs=eidxs(sol,:,:P))[end][[1:5...,7]]) # hide +nothing #hide + +# Through the magic of symbolic indexing we can plot the power flows on all lines: + +plot(sol; idxs=eidxs(sol,:,:P)) + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/cascading_failure/d235e48e.svg b/previews/PR184/generated/cascading_failure/d235e48e.svg new file mode 100644 index 00000000..2f1eec27 --- /dev/null +++ b/previews/PR184/generated/cascading_failure/d235e48e.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/cascading_failure/index.html b/previews/PR184/generated/cascading_failure/index.html new file mode 100644 index 00000000..9b498821 --- /dev/null +++ b/previews/PR184/generated/cascading_failure/index.html @@ -0,0 +1,73 @@ + +Cascading Failure · NetworkDynamics
GitHub

Cascading Failure

This script reimplements the minimal example of a dynamic cascading failure described in Schäfer et al. (2018) [1]. In is an example how to use callback functions to change network parameters. In this case to disable certain lines. This script can be dowloaded as a normal Julia script here.

[1] Schäfer, B., Witthaut, D., Timme, M., & Latora, V. (2018). Dynamically induced cascading failures in power grids. Nature communications, 9(1), 1-13. https://www.nature.com/articles/s41467-018-04287-5

The system is modeled using swing equation and active power edges. The nodes are characterized by the voltage angle δ, the active power on each line is symmetric and a function of the difference between source and destination angle δ_src - δ_dst.

using NetworkDynamics
+using Graphs
+using OrdinaryDiffEqTsit5
+using DiffEqCallbacks
+using Plots
+import SymbolicIndexingInterface as SII

For the nodes we define the swing equation. State v[1] = δ, v[2] = ω. The swing equation has three parameters: p = (P_ref, I, γ) where P_ref is the power setpopint, I is the inertia and γ is the droop or damping coeficcient.

The output of the node is just the first state. g=1 is a shorthand for g=StateMask(1:1) which implements a trivial output function g which just takes the first element of the state vector.

function swing_equation(dv, v, esum, p,t)
+    P, I, γ = p
+    dv[1] = v[2]
+    dv[2] = P - γ * v[2] .+ esum[1]
+    dv[2] = dv[2] / I
+    nothing
+end
+vertex = VertexModel(f=swing_equation, g=1, sym=[:δ, :ω], psym=[:P_ref, :I=>1, :γ=>0.1])
VertexModel :VertexM PureStateMap()
+ ├─ 2 states: [δ, ω]
+ ├─ 1 output: [δ]
+ └─ 3 params: [P_ref, I=1, γ=0.1]

Lets define a simple purely active power line whose active power flow is completlye determined by the connected voltage angles and the coupling constant K. We give an additonal parameter, the line limit, which we'll use later in the callback.

function simple_edge(e, v_s, v_d, (K,), t)
+    e[1] = K * sin(v_s[1] - v_d[1])
+end
+edge = EdgeModel(;g=AntiSymmetric(simple_edge), outsym=:P, psym=[:K=>1.63, :limit=>1])
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[₋P] dst=[P]
+ └─   2 params:  [K=1.63, limit=1]

With the definition of the graph topology we can build the Network object:

g = SimpleGraph([0 1 1 0 1;
+                 1 0 1 1 0;
+                 1 1 0 1 0;
+                 0 1 1 0 1;
+                 1 0 0 1 0])
+swing_network = Network(g, vertex, edge)
Network with 10 states and 29 parameters
+ ├─ 5 vertices (1 unique type)
+ └─ 7 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

For the parameters, we create the NWParameter object prefilled with default p values

p = NWParameter(swing_network)
+# vertices 1, 3 and 4 act as loads
+p.v[(1,3,4), :P_ref] .= -1
+# vertices 2 and 5 act as generators
+p.v[(2,5), :P_ref] .= 1.5

We can use find_fixpoint to find a valid initial condition of the network

u0 = find_fixpoint(swing_network, p)

In order to implement the line failures, we need to create a VectorContinousCallback. In the callback, we compare the current flow on the line with the limit. If the limit is reached, the coupling K is set to 0.

First we can define the affect function:

function affect!(integrator, idx)
+    println("Line $idx tripped at t=$(integrator.t)")
+    p = NWParameter(integrator) # get indexable parameter object
+    p.e[idx, :K] = 0
+    auto_dt_reset!(integrator)
+    save_parameters!(integrator)
+    nothing
+end

There is one important aspect to this function: the save_parameters! call. In the callback, we change the parameters of the network, making the parameters time dependent. The flow on the line is a function P(t) = f(u(t), p(t)). Thus we need to inform the integrator, that a discrete change in parameters happend. With this, the solution object not only tracks u(t) but also p(t) and we may extract the observable P(t) directly.

The callback trigger condition is a bit more complicated. The straight forward version looks like this:

function naive_condition(out, u, t, integrator)
+    # careful,  u != integrator.u
+    # therefore construct nwstate with Network info from integrator but u
+    s = NWState(integrator, u, integrator.p, t)
+    for i in eachindex(out)
+        out[i] = abs(s.e[i,:P]) - s.p.e[1,:limit] # compare flow with limit for line
+    end
+    nothing
+end

However, from a performacne perspectiv there are problems with this solution: on every call, we need to perform symbolic indexing into the NWState object. Symbolic indexing is not cheap, as it requires to gather meta data about the network. Luckily, the SymbolicIndexingInterface package which powers the symbolic indexing provides the lower level functions getp and getu which can be used to create and cache accessors to the internal states.

This still isn't ideal beacuse both getlim and getflow getters will create arrays within the callback. But is far better then resolving the flat state indices every time.

condition = let getlim = SII.getp(swing_network, epidxs(swing_network, :, :limit)),
+                getflow = SII.getu(swing_network, eidxs(swing_network, :, :P))
+    function (out, u, t, integrator)
+        # careful,  u != integrator.u
+        # therefore construct nwstate with Network info from integrator but u
+        s = NWState(integrator, u, integrator.p, t)
+        out .= getlim(s) .- abs.(getflow(s))
+        nothing
+    end
+end

We can combine affect and condition to form the callback.

trip_cb = VectorContinuousCallback(condition, affect!, ne(g));

However, there is another component missing. If we look at the powerflow on the lines in the initial steady state

u0.e[:, :P]
7-element Vector{Float64}:
+ -0.44001353955110956
+  0.19001353955110994
+ -0.7500000000000001
+  0.6199729208977807
+  0.4400135395511094
+ -0.1900135395511101
+ -0.75

We see that every flow is below the trip value 1.0. Therefor we need to add a distrubance to the network. We do this by manually disabeling line 5 at time 1.

trip_first_cb = PresetTimeCallback(1.0, integrator->affect!(integrator, 5));

With those components, we can create the problem and solve it.

prob = ODEProblem(swing_network, uflat(u0), (0,6), copy(pflat(p));
+                  callback=CallbackSet(trip_cb, trip_first_cb))
+sol = solve(prob, Tsit5());
Line 5 tripped at t=1.0
+Line 7 tripped at t=2.247676397005474
+Line 4 tripped at t=2.502523192233235
+Line 1 tripped at t=3.1947647115093654
+Line 3 tripped at t=3.3380530127462587
+Line 2 tripped at t=3.4042696241577888

Through the magic of symbolic indexing we can plot the power flows on all lines:

plot(sol; idxs=eidxs(sol,:,:P))
Example block output

This page was generated using Literate.jl.

diff --git a/previews/PR184/generated/directed_and_weighted_graphs.jl b/previews/PR184/generated/directed_and_weighted_graphs.jl new file mode 100644 index 00000000..1a989b8b --- /dev/null +++ b/previews/PR184/generated/directed_and_weighted_graphs.jl @@ -0,0 +1,121 @@ +# # Neurodynamic model of synchronization in the human brain +# +# +# #### Topics covered in this tutorial include: +# +# - constructing a directed, weighted graph from data +# - some useful macros +# - parameter handling +# - stiff equations +# +# ## The FitzHugh-Nagumo model +# +# Dynamics of spiking neurons have been described in a simplified manner by the [FitzHugh-Nagumo model](https://en.wikipedia.org/wiki/FitzHugh%E2%80%93Nagumo_model). +# +# ```math +# \begin{aligned} +# \varepsilon \dot u & = u - \frac{u^3}{3} - v \\ +# \dot v & = u + a +# \end{aligned} +# ``` +# +# Here $u$ is a fast, excitatory variable corresponding to the membrane potential and $v$ is a slower, inhibitory varibale. $\varepsilon$ is a parameter separating these time-scales, and $a$ is a control parameter. +# +# In simplified models of the brain, such *relaxation oscillators* may be used to model individual neurons, clusters of neurons or even larger areas in the brain. The FitzHugh-Nagumo model has been widely used for studying synchronization in neuronal activity, which in turn has been connected to physiological phenomena such as epileptic seizures. +# +# ## Coupling relaxation oscillators +# +# While different coupling schemes for FitzHugh-Nagumo oscillators have been proposed, in this tutorial we focus on coupling of the excitatory variables via electrical gap junctions, as described by the following system of equations. +# +# ```math +# \begin{aligned} +# \varepsilon \dot u_i & = u_i - \frac{u_i^3}{3} - v_i - \sigma \sum_{j=1}^N G_{ij}(u_i - u_j) \\ +# \dot v_i & = u_i + a +# \end{aligned} +# ``` +# +# This is a simple diffusive coupling mediated by the difference between activation potentials in pairs of neurons. A similar coupling term was introduced in the "getting started" tutorial. +# +# ## The network topology - a brain atlas +# +# In the following we will use a directed and weighted network encoding the strength and directionality of coupling between 90 different areas of the brain [Nathalie Tzourio-Mazoyer et al., 2002, Neuroimage]. +# +# The network weight matrix is given as a text file containing 90 lines with 90 numbers representing the coupling strength and separated by commas `,`. The data can be conveniently read into a matrix with the `DelimitedFiles` module. + +using DelimitedFiles +using SimpleWeightedGraphs, Graphs +using NetworkDynamics +using OrdinaryDiffEqTsit5 +using OrdinaryDiffEqSDIRK +using StableRNGs +using Plots + +# adjust the load path for your filesystem! +file = joinpath(pkgdir(NetworkDynamics), "docs", "examples", "Norm_G_DTI.txt") +G = readdlm(file, ',', Float64, '\n') + +# The data structure for directed, weighted graphs is provided by the package `SimpleWeightedGraphs.jl` which is based on `Graphs.jl`. + +# First we construct a weighted, directed graph +g_weighted = SimpleWeightedDiGraph(G) + +# For later use we extract the edge.weight attributes +# . is the broadcasting operator and gets the attribute :weight for every edge +edge_weights = getfield.(collect(edges(g_weighted)), :weight) + +# we promote the g_weighted graph as a directed graph (weights of the edges are included in parameters) +g_directed = SimpleDiGraph(g_weighted) + +# ## Setting up the ODEProblem +# +# Defining the `VertexModel` and `EdgeModel` is similar to the example before. The macro `Base.@propagate_inbounds` tells the compiler to inline the function and propagate the inbounds context. For more details see the julia [documentation](https://docs.julialang.org/en/v1/devdocs/boundscheck/). + +Base.@propagate_inbounds function fhn_electrical_vertex!(dv, v, esum, p, t) + (a, ϵ) = p + dv[1] = v[1] - v[1]^3 / 3 - v[2] + esum[1] + dv[2] = (v[1] - a) * ϵ + nothing +end +vertex = VertexModel(f=fhn_electrical_vertex!, g=1, sym=[:u, :v], psym=[:a=>0.5, :ϵ=>0.05]) + +Base.@propagate_inbounds function electrical_edge!(e, v_s, v_d, (w, σ), t) + e[1] = w * (v_s[1] - v_d[1]) * σ + nothing +end +electricaledge = EdgeModel(g=Directed(electrical_edge!), outdim=1, psym=[:weight, :σ=>0.5]) + +fhn_network! = Network(g_directed, vertex, electricaledge) + +# Since this system is a directed one with thus directed edges, the keyword argument `coupling` is used to set the coupling of the edges to `Directed()`. +# +# ## Parameter handling +# +# Some of the parameters have been declared with default values. +# Those default values will be used when creating the `NWParameter` object. +# We can use getindex on the parameter objects to set the missing weight values. + +p = NWParameter(fhn_network!) +p.e[1:ne(g_directed), :weight] = edge_weights + +# The initial conditions could be created similarly to the parameters as an indexable `NWState` obejct. +# Since we chose a random initial condition we initialize the flat array directly: + +x0 = randn(StableRNG(42), dim(fhn_network!)) * 5 + +# ## Solving the system +# +# Now we are ready to create an `ODEProblem`. Since for some choices of parameters the FitzHugh-Nagumo model is *stiff* (i.e. numerically unstable), we use a solver with automated stiffness detection. Such a solver switches to a more stable solver only when the solution enters a region of phase space where the problem is numerically unstable. In this case we use `Tsit5` and switch to `TRBDF2` when necessary. `AutoTsit5` is the switching version of the `Tsit5` algorithm. +# +# Not that we call `pflat` on the `NWParameter` object to get the flat array of parameters. + +tspan = (0.0, 200.0) +prob = ODEProblem(fhn_network!, x0, tspan, pflat(p)) +sol = solve(prob, AutoTsit5(TRBDF2())); + +# ## Plotting +# +# The plot of the excitatory variables shows that they synchronize for this choice of parameters. + +plot(sol; idxs=vidxs(fhn_network!, :, :u), legend=false, ylim=(-5, 5), fmt=:png) + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/directed_and_weighted_graphs/e002d24b.svg b/previews/PR184/generated/directed_and_weighted_graphs/e002d24b.svg new file mode 100644 index 00000000..5e035eea --- /dev/null +++ b/previews/PR184/generated/directed_and_weighted_graphs/e002d24b.svg @@ -0,0 +1,132 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/directed_and_weighted_graphs/index.html b/previews/PR184/generated/directed_and_weighted_graphs/index.html new file mode 100644 index 00000000..c06a5b52 --- /dev/null +++ b/previews/PR184/generated/directed_and_weighted_graphs/index.html @@ -0,0 +1,48 @@ + +Directed and Weighted Graphs · NetworkDynamics
GitHub

Neurodynamic model of synchronization in the human brain

This example can be dowloaded as a normal Julia script here.

Topics covered in this tutorial include:

  • constructing a directed, weighted graph from data
  • some useful macros
  • parameter handling
  • stiff equations

The FitzHugh-Nagumo model

Dynamics of spiking neurons have been described in a simplified manner by the FitzHugh-Nagumo model.

\[\begin{aligned} +\varepsilon \dot u & = u - \frac{u^3}{3} - v \\ +\dot v & = u + a +\end{aligned}\]

Here $u$ is a fast, excitatory variable corresponding to the membrane potential and $v$ is a slower, inhibitory varibale. $\varepsilon$ is a parameter separating these time-scales, and $a$ is a control parameter.

In simplified models of the brain, such relaxation oscillators may be used to model individual neurons, clusters of neurons or even larger areas in the brain. The FitzHugh-Nagumo model has been widely used for studying synchronization in neuronal activity, which in turn has been connected to physiological phenomena such as epileptic seizures.

Coupling relaxation oscillators

While different coupling schemes for FitzHugh-Nagumo oscillators have been proposed, in this tutorial we focus on coupling of the excitatory variables via electrical gap junctions, as described by the following system of equations.

\[\begin{aligned} +\varepsilon \dot u_i & = u_i - \frac{u_i^3}{3} - v_i - \sigma \sum_{j=1}^N G_{ij}(u_i - u_j) \\ +\dot v_i & = u_i + a +\end{aligned}\]

This is a simple diffusive coupling mediated by the difference between activation potentials in pairs of neurons. A similar coupling term was introduced in the "getting started" tutorial.

The network topology - a brain atlas

In the following we will use a directed and weighted network encoding the strength and directionality of coupling between 90 different areas of the brain [Nathalie Tzourio-Mazoyer et al., 2002, Neuroimage].

The network weight matrix is given as a text file containing 90 lines with 90 numbers representing the coupling strength and separated by commas ,. The data can be conveniently read into a matrix with the DelimitedFiles module.

using DelimitedFiles
+using SimpleWeightedGraphs, Graphs
+using NetworkDynamics
+using OrdinaryDiffEqTsit5
+using OrdinaryDiffEqSDIRK
+using StableRNGs
+using Plots
+
+# adjust the load path for your filesystem!
+file = joinpath(pkgdir(NetworkDynamics), "docs", "examples", "Norm_G_DTI.txt")
+G = readdlm(file, ',', Float64, '\n')

The data structure for directed, weighted graphs is provided by the package SimpleWeightedGraphs.jl which is based on Graphs.jl.

# First we construct a weighted, directed graph
+g_weighted = SimpleWeightedDiGraph(G)
+
+# For later use we extract the edge.weight attributes
+# . is the broadcasting operator and gets the attribute :weight for every edge
+edge_weights = getfield.(collect(edges(g_weighted)), :weight)
+
+# we promote the g_weighted graph as a directed graph (weights of the edges are included in parameters)
+g_directed = SimpleDiGraph(g_weighted)

Setting up the ODEProblem

Defining the VertexModel and EdgeModel is similar to the example before. The macro Base.@propagate_inbounds tells the compiler to inline the function and propagate the inbounds context. For more details see the julia documentation.

Base.@propagate_inbounds function fhn_electrical_vertex!(dv, v, esum, p, t)
+    (a, ϵ) = p
+    dv[1] = v[1] - v[1]^3 / 3 - v[2] + esum[1]
+    dv[2] = (v[1] - a) * ϵ
+    nothing
+end
+vertex = VertexModel(f=fhn_electrical_vertex!, g=1, sym=[:u, :v], psym=[:a=>0.5, :ϵ=>0.05])
VertexModel :VertexM PureStateMap()
+ ├─ 2 states: [u, v]
+ ├─ 1 output: [u]
+ └─ 2 params: [a=0.5, ϵ=0.05]
Base.@propagate_inbounds function electrical_edge!(e, v_s, v_d, (w, σ), t)
+    e[1] = w * (v_s[1] - v_d[1]) * σ
+    nothing
+end
+electricaledge = EdgeModel(g=Directed(electrical_edge!), outdim=1, psym=[:weight, :σ=>0.5])
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ ├─ 0/1 outputs: src=[] dst=[o]
+ └─   2 params:  [weight, σ=0.5]
fhn_network! = Network(g_directed, vertex, electricaledge)
Network with 180 states and 15766 parameters
+ ├─ 90 vertices (1 unique type)
+ └─ 7793 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

Since this system is a directed one with thus directed edges, the keyword argument coupling is used to set the coupling of the edges to Directed().

Parameter handling

Some of the parameters have been declared with default values. Those default values will be used when creating the NWParameter object. We can use getindex on the parameter objects to set the missing weight values.

p = NWParameter(fhn_network!)
+p.e[1:ne(g_directed), :weight] = edge_weights

The initial conditions could be created similarly to the parameters as an indexable NWState obejct. Since we chose a random initial condition we initialize the flat array directly:

x0 = randn(StableRNG(42), dim(fhn_network!)) * 5

Solving the system

Now we are ready to create an ODEProblem. Since for some choices of parameters the FitzHugh-Nagumo model is stiff (i.e. numerically unstable), we use a solver with automated stiffness detection. Such a solver switches to a more stable solver only when the solution enters a region of phase space where the problem is numerically unstable. In this case we use Tsit5 and switch to TRBDF2 when necessary. AutoTsit5 is the switching version of the Tsit5 algorithm.

Not that we call pflat on the NWParameter object to get the flat array of parameters.

tspan = (0.0, 200.0)
+prob  = ODEProblem(fhn_network!, x0, tspan, pflat(p))
+sol = solve(prob, AutoTsit5(TRBDF2()));

Plotting

The plot of the excitatory variables shows that they synchronize for this choice of parameters.

plot(sol; idxs=vidxs(fhn_network!, :, :u), legend=false, ylim=(-5, 5), fmt=:png)
Example block output

This page was generated using Literate.jl.

diff --git a/previews/PR184/generated/gas_network.jl b/previews/PR184/generated/gas_network.jl new file mode 100644 index 00000000..4385ea1d --- /dev/null +++ b/previews/PR184/generated/gas_network.jl @@ -0,0 +1,379 @@ +# # Dynamic Flow in simple Gas Network +# +# This Example is based on the paper +# +# > Albertus J. Malan, Lukas Rausche, Felix Strehle, Sören Hohmann, +# > Port-Hamiltonian Modelling for Analysis and Control of Gas Networks, +# > IFAC-PapersOnLine, Volume 56, Issue 2, 2023, https://doi.org/10.1016/j.ifacol.2023.10.193. +# +# and tries replicate a simple simulation of flow in a 3-node gas network. +# +# +# We start by importing the necessary packages: + +using NetworkDynamics +using ModelingToolkit +using DynamicQuantities +using ModelingToolkit: D as Dt, t as t +using Test +using StaticArrays +using LinearInterpolations +using OrdinaryDiffEqTsit5 +using CairoMakie +CairoMakie.activate!(type="svg") #hide + +# ## Node Models +# +# In this example, we use the equation based modeling using `ModelingToolkit.jl`. +# To verify the equations on a basic level we also provide units to eveything to +# perform dimensionality checks. +# +# There are 2 node models used in the paper. The first node type has a constant +# pressure. +# Additionally, we ad some "internal" state `q̃_inj` which we want to plot later (see also Observables). + +@mtkmodel ConstantPressureNode begin + @parameters begin + p_set, [description="Constant pressure setpoint", unit=u"Pa"] + end + @variables begin + p(t) = p_set, [description="Pressure", unit=u"Pa", output=true] + q̃_nw(t), [description="aggregated flow from pipes into node", unit=u"m^3/s", input=true] + q̃_inj(t), [description="internal state for introspection", unit=u"m^3/s"] + end + @equations begin + p ~ p_set + q̃_inj ~ -q̃_nw + end +end +nothing #hide + +# The second node model is a variable pressure node. It has one output state (the pressure) and one input state, +# the aggregated flows from the connected pipes. +# As an internal state we have the injected flow from our source/load. +# The source/load behaviour itself is provided via a time dependent function. + +@mtkmodel VariablePressureNode begin + @structural_parameters begin + load_profile # time dependent load profile + end + @parameters begin + C, [description="Lumped capacitance of connected pipes", unit=u"m^4 * s^2 / kg"] + end + @variables begin + p(t)=5e6, [description="Pressure", unit=u"Pa", output=true] + q̃_inj(t), [description="external injection into node", unit=u"m^3/s"] + q̃_nw(t), [description="aggregated flow from pipes into node", unit=u"m^3/s", input=true] + end + @equations begin + q̃_inj ~ load_profile(t) + C * Dt(p) ~ q̃_inj + q̃_nw # (30) + end +end +nothing #hide + +# ## Pipe Model +# +# The pipe is modeld as a first order ODE for the volumetric flow at the `dst` end. It has two inputs: +# the pressure at the source and and the pressure at the destination end. +# Later on, we'll specify the model to be antisymmetric, thus the flow is calculated explicitly for the +# destination end, but the source end will just recive just that times (-1). + +@mtkmodel Pipe begin + @parameters begin + L, [description="Length of pipe", unit=u"m"] + D, [description="Diameter of pipe", unit=u"m"] + A, [description="Cross-sectional area of pipe", unit=u"m^2"] + sinθ, [description="Angle of inclination" ] + γ, [description="Friction efficiency factor"] + η, [description="Dynamic viscosity", unit=u"kg/(m*s)"] + r, [description="Pipe roughness", unit=u"m"] + g, [description="Gravitational acceleration", unit=u"m/s^2"] + T, [description="simulation temperature", unit=u"K"] + Tc, [description="crictical temperature", unit=u"K"] + pc, [description="critical pressure", unit=us"Pa"] + Rs, [description="Specific gas constant for natural gas", unit=us"J/(kg*K)"] + c̃, [description="Speed of sound in fluid at standard conditions", unit=u"m/s"] + ρ̃, [description="standard density", unit=u"kg/m^3"] + p̃, [description="standard pressure", unit=us"Pa"] + end + @variables begin + p_src(t), [description="Pressure at source end", unit=us"Pa", input=true] + p_dst(t), [description="Pressure at destination end", unit=us"Pa", input=true] + q̃(t)=1, [description="Flow through pipe", unit=u"m^3/s", output=true] + Re(t), [description="Reynolds number"] + λ(t), [description="Friction factor"] + λe(t), [description="Effective friction factor"] + pM(t), [description="mean pressure", unit=us"Pa"] + Z(t), [description="compressibility factor"] + ρ(t), [description="density", unit=u"kg/m^3"] + c(t), [description="speed of sound", unit=u"m/s"] + end + @equations begin + Z ~ 1 - 3.52 * pM/pc * exp(-2.26*(T/Tc)) + 0.274 * (pM/pc)^2 * exp(-1.878*(T/Tc)) # (5) + ρ ~ pM / (Rs * T * Z) # (4) + + # TODO: Whats the correct speed of sound? + c ~ sqrt(T * Rs * Z) # (4) # pressure/temp dependent + # c ~ c̃ # "standard" speed of sound based on standard conditions + + # TODO: Whats the correct Reynolds number? + Re ~ (ρ * abs(q̃*p̃/pM) * D) / (η * A) # (6) # based "actual" conditions + # Re ~ (ρ̃ * abs(q̃) * D) / (η * A) # (6) # based on standard conditions + + λ ~ ifelse(Re < 2300, + 64/Re, # laminar (7) + (2*log10(4.518/Re * log10(Re/7) + r/(3.71*D)))^(-2) # turbulent (8) + ) + λe ~ λ/γ^2 # (10) + pM ~ 2/3*(p_src + p_dst - (p_src*p_dst)/(p_src + p_dst)) # (20) + + Dt(q̃) ~ A/(L*ρ̃)*(-(λe * ρ̃^2 * c^2 * L * abs(q̃))/(2 * D * A^2 * pM) * q̃ - (g * L * sinθ)/(c^2) * pM + (p_src - p_dst)) # (31) + end +end +nothing #hide + +# ## Parametrization +# +# The parameterization turned out to be a bit tricky. There might be errors in there. +# +# Some of them are quite cleare and explicitly given. + +g = 9.81u"m/s^2" # that we just know +Rs = 518.28u"J/(kg*K)" # Specific gas constant for natural gas +η = 1e-5u"kg/(m*s)" # Dynamic viscosity +pc = 46.5u"bar" # Critical pressure +p̃ = 1.01325u"bar" # standard pressure +Tc = 190.55u"K" # critical temperature +T̃ = 273.15u"K" # standard temperature +T = 278u"K" # simulation temperature +γ = 0.98 # friction efficiency factor +r = 0.012u"mm" # pipe roughness +D = 0.6u"m" # pipe diameter + +# TODO: here is switched the lenths l12 and l13. The results are better. Is this a mistake in the paper? +L₁₂ = 90u"km" +L₁₃ = 80u"km" +L₂₃ = 100u"km" +Δh₁ = 0u"km" # this value is different for different sims in the paper +p₁_set = 50u"bar" +nothing # hide + +# The geometric parameters for the pipes can be directly derived. + +A = π/4 * D^2 +sinθ₁₂ = ustrip(Δh₁ / L₁₂) +sinθ₁₃ = ustrip(Δh₁ / L₁₃) +sinθ₂₃ = 0.0 +nothing # hide + +# Lastly, we need to calculate the compressibility factor, the speed of sound and +# the density at standard conditions: + +Z̃ = 1 - 3.52 * p̃/pc * exp(-2.26*(T̃/Tc)) + 0.274 * (p̃/pc)^2 * exp(-1.878*(T̃/Tc)) # (5) +c̃ = sqrt(T̃ * Rs * Z̃) # (4) at standard conditions +ρ̃ = p̃ / (Rs * T̃ * Z̃) # (4) at standard conditions + +nothing # hide + +# The equivalent "pressure capacity" at the nodes is calculated as a sum of the connected +# pipe parameters according to (28). +# +# Here use defintions based on the speed and "standard" conditions. + +C₂ = L₁₂*A/(2*ρ̃*c̃^2) + L₂₃*A/(2*ρ̃*c̃^2) # (28) +C₃ = L₁₃*A/(2*ρ̃*c̃^2) + L₂₃*A/(2*ρ̃*c̃^2) # (28) +nothing #hide + +# Alternatively, we could calculate `Z2` and `Z3` based on the actuel pressure and simulation temperature. +# Then we could calculated the speed of sound for the "correct" conditions at the node. +# It seems to have very little effect on the actual results so I kept it simple. + +nothing #hide + +# ## Load Profile +# +# The paper specifies the load profile at two nodes. We use the package [`LinearInterpolations.jl`](https://github.com/jw3126/LinearInterpolations.jl) +# to get a callable object which represents this picewise linear interpolation. +# +# However, this function is not Symbolics.jl compatible, so we need to stop Symbolics.jl/ModelingToolkit.jl +# from tracing it. To do so, we use `@register_symbolic` to declare it as a symbolic function which is treated +# as a blackbox. +# +# Additionally, we need to tell ModelingToolkit about the units of this object. This is just used +# for the static unit check during construction of the model. Later one, when we generate the +# Julia code from the symbolic reepresentation all units will be stripped. +# +# > **Discontinuities in RHS** +# > +# > The picewise linear interpolated function creates discontinuities in the RHS of the system. +# > However since we know the times exactly, we can handle this by simply giving a list of explicit +# > tstops to the solve command, to make sure those are hit exactly. + +load2(t) = -Interpolate(SA[0, 4, 12, 20, 24]*3600, SA[20, 30, 10, 30, 20], extrapolate=LinearInterpolations.Constant(20))(t) +load3(t) = -Interpolate(SA[0, 4, 12, 20, 24]*3600, SA[40, 50, 30, 50, 40], extrapolate=LinearInterpolations.Constant(40))(t) +@register_symbolic load2(t) +@register_symbolic load3(t) +ModelingToolkit.get_unit(op::typeof(load2), _) = u"m^3/s" +ModelingToolkit.get_unit(op::typeof(load3), _) = u"m^3/s" +nothing #hide + +# ## Building the Network +# +# To bild the Network we first need to define the components. This is a two step process: +# +# - first create the symbolic `ODESystem` using ModelingToolkit +# - secondly build a NetworkDynamics component model (`VertexModel`/`EdgeModel`) based on the symbolic system. +# +# In the first step we can use the keyword arguments to pass "default" values for our parameters and states. +# Those values will be automaticially transfered to the metadata of the component model the second step. +# +# The second step requires to define the interface variables, i.e. what are the "input" states of your +# component model and what are the "output" states. +# For `VertexModel` the input state is the aggregated flow of all connected pipes. The output state is the pressure of the node. + +@named v1_mtk = ConstantPressureNode(p_set=p₁_set) +v1 = VertexModel(v1_mtk, [:q̃_nw], [:p]; name=:v1, vidx=1) + +@named v2_mtk = VariablePressureNode(C=C₂, load_profile=load2) +v2 = VertexModel(v2_mtk, [:q̃_nw], [:p]; name=:v2, vidx=2) + +@named v3_mtk = VariablePressureNode(C=C₃, load_profile=load3) +v3 = VertexModel(v3_mtk, [:q̃_nw], [:p]; name=:v3, vidx=3) + +# For the edge Model we have two inputs: the pressure on both source and destination end. +# There is a single output state: the volumetric flow. However we also need to tell +# NetworkDynamics about the coupling type. In this case we use `AntiSymmetric`, which +# meas that the source end will recieve the same flow, just inverted sign. + +@named e12_mtk = Pipe(; L=L₁₂, sinθ=sinθ₁₂, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃) +@named e13_mtk = Pipe(; L=L₁₃, sinθ=sinθ₁₃, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃) +@named e23_mtk = Pipe(; L=L₂₃, sinθ=sinθ₂₃, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃) + +e12 = EdgeModel(e12_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e12, src=1, dst=2) +e13 = EdgeModel(e13_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e13, src=1, dst=3) +e23 = EdgeModel(e23_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e23, src=2, dst=3) + +# To build the network object we just need to pass the vertices and edges to the constructor. +# +# Note that we've used the `vidx` and `src`/`dst` keywords in the constructors to define for +# each component to which "part" of the network it belongs. +# +# This means, the constructor can automaticially construct a graph based on those informations and +# we don't need to pass it explicitly. + +nw = Network([v1, v2, v3], [e12, e13, e23]) + +# As a result, we recive a network with 3 unique types (v2 and v3 are similar but +# structurally different, because both functions capure a unique loadprofile function). +# +# ## Finding a Steady State +# +# To simulate the systme, we first need to find a steadystate. As a "guess" for that +# we create a `NWState` object from the network. +# This will allocate flat arrays for states `u` and parameters `p` and fill them with the +# default values. + +uguess = NWState(nw) + +# This is not a steadystate of the system however. To find a true steadystate we want to ensure +# that the lhs of the system is zero. +# We can solve for a steady state numerically by defining a Nonlinear Rootfind problem. +# +# To do so, we need to wrap the Network object in a closure. + +nwwrap = (du, u, p) -> begin + nw(du, u, p, 0) + nothing +end +initprob = NonlinearProblem(nwwrap, uflat(uguess), pflat(uguess)) +initsol = solve(initprob) + +# We can create a new `NWState` object by wrapping the solution from the nonlinear problem and the +# original prameters in a new `NWState` object. + +u0 = NWState(nw, initsol.u, uguess.p) + +# ## Solving the ODE +# +# Using this as our initial state we can create the actual `ODEProblem`. +# Since the ode allways operates on flat state and aprameter arrays we use `uflat` and `pflat` to extract them. + +prob = ODEProblem(nw, uflat(u0), (0.0,24*3600), copy(pflat(u0))) +sol = solve(prob, Tsit5(), tstops=[0,4,12,20,24]*3600) +nothing #hide + +# ## Inspect the Solution +# +# Inspecting the solution is all which is left to do. + +xticks = ((0:4:24)*3600, string.(0:4:24)) # its nice to display hours +fig = begin + _fig = Figure() + row = 1 + ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="pressure [Pa]", title="Pressure at nodes", xticks) + xlims!(ax, sol.t[begin], sol.t[end]) + ylims!(ax, 47.9e5, 49.9e5) + for i in 1:3 + lines!(ax, sol, idxs=vidxs(nw, i, :p); label="v$i", color=Cycled(i)) + end + axislegend(ax) + row += 1 + + ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="flow [m³/s]", title="Flow through pipes", xticks) + xlims!(ax, sol.t[begin], sol.t[end]) + ylims!(ax, 16, 44) + for i in 1:2 + lines!(ax, sol, idxs=eidxs(nw, i, :q̃); label="e$i flow", color=Cycled(i)) + end + axislegend(ax, position=:rb) + row += 1 + _fig +end + +# Notably, the "internal" states defined in the symbolic models are not "states" in the sense of the ODE. +# For example, we captured the load profile in the `q̃_inj` state of the `VariablePressureNode`. +# The only dynamic state of the model however is `p`. +# Using the "observables" mechanism from SciML, which is implemented by NetworkDynamics, we can reconstruct +# those "optimized" states which have been removed symbolicially. +# Here we plot the reconstructed load profile of nodes 2 and 3. Also, we know that node 1 is infinetly stiff, +# acting as an infinite source of volumetric flow. We can reconstruct this flow too. + +fig = begin + _fig = Figure() + row = 1 + ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="flow [m³/s]", title="Flow at nodes", xticks) + xlims!(ax, sol.t[begin], sol.t[end]) + lines!(ax, sol, idxs=vidxs(nw, 1, :q̃_inj); label="v1 compensation", color=Cycled(1)) + for i in 2:3 + lines!(ax, sol, idxs=vidxs(nw, i, :q̃_inj); label="v$i load profile", color=Cycled(i)) + end + axislegend(ax, position=:rc) + _fig +end + +# Lastly we want to observe two internal states of the pipes: the Reynolds number and the mean pressure. +# We see, that we're purely in the turbulent flow regime. + +fig = begin + _fig = Figure() + row = 1 + ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="Reynolds number", title="Reynolds number", xticks) + xlims!(ax, sol.t[begin], sol.t[end]) + for i in 1:3 + lines!(ax, sol, idxs=eidxs(nw, i, :Re); label="e $i", color=Cycled(i)) + end + hlines!(ax, 2300, color=:black, linestyle=:dash, label="L/T transition") + axislegend(ax, position=:rb) + row += 1 + + ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="Mean pressure [Pa]", title="Mean pressure in pipes", xticks) + xlims!(ax, sol.t[begin], sol.t[end]) + for i in 1:3 + lines!(ax, sol, idxs=eidxs(nw, i, :pM); label="e $i", color=Cycled(i)) + end + axislegend(ax, position=:rb) + _fig +end + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/gas_network/43f744c5.svg b/previews/PR184/generated/gas_network/43f744c5.svg new file mode 100644 index 00000000..fc1b195b --- /dev/null +++ b/previews/PR184/generated/gas_network/43f744c5.svg @@ -0,0 +1,1008 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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GitHub

Dynamic Flow in simple Gas Network

This Example is based on the paper

Albertus J. Malan, Lukas Rausche, Felix Strehle, Sören Hohmann, Port-Hamiltonian Modelling for Analysis and Control of Gas Networks, IFAC-PapersOnLine, Volume 56, Issue 2, 2023, https://doi.org/10.1016/j.ifacol.2023.10.193.

and tries replicate a simple simulation of flow in a 3-node gas network.

This example can be dowloaded as a normal Julia script here.

We start by importing the necessary packages:

using NetworkDynamics
+using ModelingToolkit
+using DynamicQuantities
+using ModelingToolkit: D as Dt, t as t
+using Test
+using StaticArrays
+using LinearInterpolations
+using OrdinaryDiffEqTsit5
+using CairoMakie

Node Models

In this example, we use the equation based modeling using ModelingToolkit.jl. To verify the equations on a basic level we also provide units to eveything to perform dimensionality checks.

There are 2 node models used in the paper. The first node type has a constant pressure. Additionally, we ad some "internal" state q̃_inj which we want to plot later (see also Observables).

@mtkmodel ConstantPressureNode begin
+    @parameters begin
+        p_set, [description="Constant pressure setpoint", unit=u"Pa"]
+    end
+    @variables begin
+        p(t) = p_set, [description="Pressure", unit=u"Pa", output=true]
+        q̃_nw(t), [description="aggregated flow from pipes into node", unit=u"m^3/s", input=true]
+        q̃_inj(t), [description="internal state for introspection", unit=u"m^3/s"]
+    end
+    @equations begin
+        p ~ p_set
+        q̃_inj ~ -q̃_nw
+    end
+end

The second node model is a variable pressure node. It has one output state (the pressure) and one input state, the aggregated flows from the connected pipes. As an internal state we have the injected flow from our source/load. The source/load behaviour itself is provided via a time dependent function.

@mtkmodel VariablePressureNode begin
+    @structural_parameters begin
+        load_profile # time dependent load profile
+    end
+    @parameters begin
+        C, [description="Lumped capacitance of connected pipes", unit=u"m^4 * s^2 / kg"]
+    end
+    @variables begin
+        p(t)=5e6, [description="Pressure", unit=u"Pa", output=true]
+        q̃_inj(t), [description="external injection into node", unit=u"m^3/s"]
+        q̃_nw(t), [description="aggregated flow from pipes into node", unit=u"m^3/s", input=true]
+    end
+    @equations begin
+        q̃_inj ~ load_profile(t)
+        C * Dt(p) ~ q̃_inj + q̃_nw # (30)
+    end
+end

Pipe Model

The pipe is modeld as a first order ODE for the volumetric flow at the dst end. It has two inputs: the pressure at the source and and the pressure at the destination end. Later on, we'll specify the model to be antisymmetric, thus the flow is calculated explicitly for the destination end, but the source end will just recive just that times (-1).

@mtkmodel Pipe begin
+    @parameters begin
+        L, [description="Length of pipe", unit=u"m"]
+        D, [description="Diameter of pipe", unit=u"m"]
+        A, [description="Cross-sectional area of pipe", unit=u"m^2"]
+        sinθ, [description="Angle of inclination" ]
+        γ, [description="Friction efficiency factor"]
+        η, [description="Dynamic viscosity", unit=u"kg/(m*s)"]
+        r, [description="Pipe roughness", unit=u"m"]
+        g, [description="Gravitational acceleration", unit=u"m/s^2"]
+        T, [description="simulation temperature", unit=u"K"]
+        Tc, [description="crictical temperature", unit=u"K"]
+        pc, [description="critical pressure", unit=us"Pa"]
+        Rs, [description="Specific gas constant for natural gas", unit=us"J/(kg*K)"]
+        c̃, [description="Speed of sound in fluid at standard conditions", unit=u"m/s"]
+        ρ̃, [description="standard density", unit=u"kg/m^3"]
+        p̃, [description="standard pressure", unit=us"Pa"]
+    end
+    @variables begin
+        p_src(t), [description="Pressure at source end", unit=us"Pa", input=true]
+        p_dst(t), [description="Pressure at destination end", unit=us"Pa", input=true]
+        q̃(t)=1, [description="Flow through pipe", unit=u"m^3/s", output=true]
+        Re(t), [description="Reynolds number"]
+        λ(t), [description="Friction factor"]
+        λe(t), [description="Effective friction factor"]
+        pM(t), [description="mean pressure", unit=us"Pa"]
+        Z(t), [description="compressibility factor"]
+        ρ(t), [description="density", unit=u"kg/m^3"]
+        c(t), [description="speed of sound", unit=u"m/s"]
+    end
+    @equations begin
+        Z ~ 1 - 3.52 * pM/pc * exp(-2.26*(T/Tc)) + 0.274 * (pM/pc)^2 * exp(-1.878*(T/Tc)) # (5)
+        ρ ~ pM / (Rs * T * Z) # (4)
+
+        # TODO: Whats the correct speed of sound?
+        c ~ sqrt(T * Rs * Z) # (4) # pressure/temp dependent
+        # c ~ c̃                   # "standard" speed of sound based on standard conditions
+
+        # TODO: Whats the correct Reynolds number?
+        Re ~ (ρ * abs(q̃*p̃/pM) * D) / (η * A) # (6) # based "actual" conditions
+        # Re ~ (ρ̃ * abs(q̃) * D) / (η * A) # (6)   # based on standard conditions
+
+        λ ~ ifelse(Re < 2300,
+            64/Re, # laminar (7)
+            (2*log10(4.518/Re * log10(Re/7) + r/(3.71*D)))^(-2) # turbulent (8)
+        )
+        λe ~ λ/γ^2 # (10)
+        pM ~ 2/3*(p_src + p_dst - (p_src*p_dst)/(p_src + p_dst)) # (20)
+
+        Dt(q̃) ~ A/(L*ρ̃)*(-(λe * ρ̃^2 * c^2 * L * abs(q̃))/(2 * D * A^2 * pM) * q̃ - (g * L * sinθ)/(c^2) * pM + (p_src - p_dst)) # (31)
+    end
+end

Parametrization

The parameterization turned out to be a bit tricky. There might be errors in there.

Some of them are quite cleare and explicitly given.

g = 9.81u"m/s^2"       # that we just know
+Rs = 518.28u"J/(kg*K)" # Specific gas constant for natural gas
+η  = 1e-5u"kg/(m*s)"   # Dynamic viscosity
+pc = 46.5u"bar"        # Critical pressure
+p̃  = 1.01325u"bar"     # standard pressure
+Tc = 190.55u"K"        # critical temperature
+T̃  = 273.15u"K"        # standard temperature
+T  = 278u"K"           # simulation temperature
+γ  = 0.98              # friction efficiency factor
+r  = 0.012u"mm"        # pipe roughness
+D  = 0.6u"m"           # pipe diameter
+
+# TODO: here is switched the lenths l12 and l13. The results are better. Is this a mistake in the paper?
+L₁₂ = 90u"km"
+L₁₃ = 80u"km"
+L₂₃ = 100u"km"
+Δh₁ = 0u"km"           # this value is different for different sims in the paper
+p₁_set = 50u"bar"

The geometric parameters for the pipes can be directly derived.

A = π/4 * D^2
+sinθ₁₂ = ustrip(Δh₁ / L₁₂)
+sinθ₁₃ = ustrip(Δh₁ / L₁₃)
+sinθ₂₃ = 0.0

Lastly, we need to calculate the compressibility factor, the speed of sound and the density at standard conditions:

Z̃ = 1 - 3.52 * p̃/pc * exp(-2.26*(T̃/Tc)) + 0.274 * (p̃/pc)^2 * exp(-1.878*(T̃/Tc)) # (5)
+c̃ = sqrt(T̃ * Rs * Z̃) # (4) at standard conditions
+ρ̃ = p̃ / (Rs * T̃ * Z̃) # (4) at standard conditions

The equivalent "pressure capacity" at the nodes is calculated as a sum of the connected pipe parameters according to (28).

Here use defintions based on the speed and "standard" conditions.

C₂ = L₁₂*A/(2*ρ̃*c̃^2) + L₂₃*A/(2*ρ̃*c̃^2) # (28)
+C₃ = L₁₃*A/(2*ρ̃*c̃^2) + L₂₃*A/(2*ρ̃*c̃^2) # (28)

Alternatively, we could calculate Z2 and Z3 based on the actuel pressure and simulation temperature. Then we could calculated the speed of sound for the "correct" conditions at the node. It seems to have very little effect on the actual results so I kept it simple.

Load Profile

The paper specifies the load profile at two nodes. We use the package LinearInterpolations.jl to get a callable object which represents this picewise linear interpolation.

However, this function is not Symbolics.jl compatible, so we need to stop Symbolics.jl/ModelingToolkit.jl from tracing it. To do so, we use @register_symbolic to declare it as a symbolic function which is treated as a blackbox.

Additionally, we need to tell ModelingToolkit about the units of this object. This is just used for the static unit check during construction of the model. Later one, when we generate the Julia code from the symbolic reepresentation all units will be stripped.

Discontinuities in RHS

The picewise linear interpolated function creates discontinuities in the RHS of the system. However since we know the times exactly, we can handle this by simply giving a list of explicit tstops to the solve command, to make sure those are hit exactly.

load2(t) = -Interpolate(SA[0, 4, 12, 20, 24]*3600, SA[20, 30, 10, 30, 20], extrapolate=LinearInterpolations.Constant(20))(t)
+load3(t) = -Interpolate(SA[0, 4, 12, 20, 24]*3600, SA[40, 50, 30, 50, 40], extrapolate=LinearInterpolations.Constant(40))(t)
+@register_symbolic load2(t)
+@register_symbolic load3(t)
+ModelingToolkit.get_unit(op::typeof(load2), _) = u"m^3/s"
+ModelingToolkit.get_unit(op::typeof(load3), _) = u"m^3/s"

Building the Network

To bild the Network we first need to define the components. This is a two step process:

  • first create the symbolic ODESystem using ModelingToolkit
  • secondly build a NetworkDynamics component model (VertexModel/EdgeModel) based on the symbolic system.

In the first step we can use the keyword arguments to pass "default" values for our parameters and states. Those values will be automaticially transfered to the metadata of the component model the second step.

The second step requires to define the interface variables, i.e. what are the "input" states of your component model and what are the "output" states. For VertexModel the input state is the aggregated flow of all connected pipes. The output state is the pressure of the node.

@named v1_mtk = ConstantPressureNode(p_set=p₁_set)
+v1 = VertexModel(v1_mtk, [:q̃_nw], [:p]; name=:v1, vidx=1)
VertexModel :v1 NoFeedForward() @ Vertex 1
+ ├─ 1 input:  [q̃_nw]
+ ├─ 0 states: []
+ ├─ 1 output: [p=5e6]
+ └─ 1 param:  [p_set=5e6]
@named v2_mtk = VariablePressureNode(C=C₂, load_profile=load2)
+v2 = VertexModel(v2_mtk, [:q̃_nw], [:p]; name=:v2, vidx=2)
VertexModel :v2 PureStateMap() @ Vertex 2
+ ├─ 1 input:  [q̃_nw]
+ ├─ 1 state:  [p=5e6]
+ ├─ 1 output: [p=5e6]
+ └─ 1 param:  [C=0.26509]
@named v3_mtk = VariablePressureNode(C=C₃, load_profile=load3)
+v3 = VertexModel(v3_mtk, [:q̃_nw], [:p]; name=:v3, vidx=3)
VertexModel :v3 PureStateMap() @ Vertex 3
+ ├─ 1 input:  [q̃_nw]
+ ├─ 1 state:  [p=5e6]
+ ├─ 1 output: [p=5e6]
+ └─ 1 param:  [C=0.25114]

For the edge Model we have two inputs: the pressure on both source and destination end. There is a single output state: the volumetric flow. However we also need to tell NetworkDynamics about the coupling type. In this case we use AntiSymmetric, which meas that the source end will recieve the same flow, just inverted sign.

@named e12_mtk = Pipe(; L=L₁₂, sinθ=sinθ₁₂, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃)
+@named e13_mtk = Pipe(; L=L₁₃, sinθ=sinθ₁₃, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃)
+@named e23_mtk = Pipe(; L=L₂₃, sinθ=sinθ₂₃, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃)
+
+e12 = EdgeModel(e12_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e12, src=1, dst=2)
+e13 = EdgeModel(e13_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e13, src=1, dst=3)
+e23 = EdgeModel(e23_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e23, src=2, dst=3)
EdgeModel :e23 PureStateMap() @ Edge 2=>3
+ ├─ 1/1 inputs:  src=[p_src] dst=[p_dst]
+ ├─   1 state:   [q̃=1]
+ ├─ 1/1 outputs: src=[₋q̃] dst=[q̃=1]
+ └─  14 params:  [ρ̃=0.71788, T=278, Rs=518.28, L=1e5, γ=0.98, D=0.6, p̃=1.0132e5, pc=4.65e6, η=1e-5, sinθ=0, Tc=190.55, A=0.28274, r=1.2e-5, g=9.81]

To build the network object we just need to pass the vertices and edges to the constructor.

Note that we've used the vidx and src/dst keywords in the constructors to define for each component to which "part" of the network it belongs.

This means, the constructor can automaticially construct a graph based on those informations and we don't need to pass it explicitly.

nw = Network([v1, v2, v3], [e12, e13, e23])
Network with 5 states and 45 parameters
+ ├─ 3 vertices (3 unique types)
+ └─ 3 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

As a result, we recive a network with 3 unique types (v2 and v3 are similar but structurally different, because both functions capure a unique loadprofile function).

Finding a Steady State

To simulate the systme, we first need to find a steadystate. As a "guess" for that we create a NWState object from the network. This will allocate flat arrays for states u and parameters p and fill them with the default values.

uguess = NWState(nw)
NWState{Vector{Float64}} of Network (3 vertices, 3 edges)
+  ├─ VIndex(2, :p)  => 5.0e6
+  ├─ VIndex(3, :p)  => 5.0e6
+  ├─ EIndex(1, :q̃)  => 1.0
+  ├─ EIndex(2, :q̃)  => 1.0
+  └─ EIndex(3, :q̃)  => 1.0
+ p = NWParameter([5.0e6, 0.265094, 0.251141, 0.717884, 278.0, 518.28, 90000.0, 0.98, 0.6, 101325.0  …  0.98, 0.6, 101325.0, 4.65e6, 1.0e-5, 0.0, 190.55, 0.282743, 1.2e-5, 9.81])
+ t = nothing

This is not a steadystate of the system however. To find a true steadystate we want to ensure that the lhs of the system is zero. We can solve for a steady state numerically by defining a Nonlinear Rootfind problem.

To do so, we need to wrap the Network object in a closure.

nwwrap = (du, u, p) -> begin
+    nw(du, u, p, 0)
+    nothing
+end
+initprob = NonlinearProblem(nwwrap, uflat(uguess), pflat(uguess))
+initsol = solve(initprob)
retcode: Success
+u: 5-element Vector{Float64}:
+  4.89510907098914e6
+  4.883573000934716e6
+ 28.277070738769
+ 31.722929261231
+  8.277070738768998

We can create a new NWState object by wrapping the solution from the nonlinear problem and the original prameters in a new NWState object.

u0 = NWState(nw, initsol.u, uguess.p)
NWState{Vector{Float64}} of Network (3 vertices, 3 edges)
+  ├─ VIndex(2, :p)  => 4.89510907098914e6
+  ├─ VIndex(3, :p)  => 4.883573000934716e6
+  ├─ EIndex(1, :q̃)  => 28.277070738769
+  ├─ EIndex(2, :q̃)  => 31.722929261231
+  └─ EIndex(3, :q̃)  => 8.277070738768998
+ p = NWParameter([5.0e6, 0.265094, 0.251141, 0.717884, 278.0, 518.28, 90000.0, 0.98, 0.6, 101325.0  …  0.98, 0.6, 101325.0, 4.65e6, 1.0e-5, 0.0, 190.55, 0.282743, 1.2e-5, 9.81])
+ t = nothing

Solving the ODE

Using this as our initial state we can create the actual ODEProblem. Since the ode allways operates on flat state and aprameter arrays we use uflat and pflat to extract them.

prob = ODEProblem(nw, uflat(u0), (0.0,24*3600), copy(pflat(u0)))
+sol = solve(prob, Tsit5(), tstops=[0,4,12,20,24]*3600)

Inspect the Solution

Inspecting the solution is all which is left to do.

xticks = ((0:4:24)*3600, string.(0:4:24)) # its nice to display hours
+fig = begin
+    _fig = Figure()
+    row = 1
+    ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="pressure [Pa]", title="Pressure at nodes", xticks)
+    xlims!(ax, sol.t[begin], sol.t[end])
+    ylims!(ax, 47.9e5, 49.9e5)
+    for i in 1:3
+        lines!(ax, sol, idxs=vidxs(nw, i, :p); label="v$i", color=Cycled(i))
+    end
+    axislegend(ax)
+    row += 1
+
+    ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="flow [m³/s]", title="Flow through pipes", xticks)
+    xlims!(ax, sol.t[begin], sol.t[end])
+    ylims!(ax, 16, 44)
+    for i in 1:2
+        lines!(ax, sol, idxs=eidxs(nw, i, :q̃); label="e$i flow", color=Cycled(i))
+    end
+    axislegend(ax, position=:rb)
+    row += 1
+    _fig
+end
Example block output

Notably, the "internal" states defined in the symbolic models are not "states" in the sense of the ODE. For example, we captured the load profile in the q̃_inj state of the VariablePressureNode. The only dynamic state of the model however is p. Using the "observables" mechanism from SciML, which is implemented by NetworkDynamics, we can reconstruct those "optimized" states which have been removed symbolicially. Here we plot the reconstructed load profile of nodes 2 and 3. Also, we know that node 1 is infinetly stiff, acting as an infinite source of volumetric flow. We can reconstruct this flow too.

fig = begin
+    _fig = Figure()
+    row = 1
+    ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="flow [m³/s]", title="Flow at nodes", xticks)
+    xlims!(ax, sol.t[begin], sol.t[end])
+    lines!(ax, sol, idxs=vidxs(nw, 1, :q̃_inj); label="v1 compensation", color=Cycled(1))
+    for i in 2:3
+        lines!(ax, sol, idxs=vidxs(nw, i, :q̃_inj); label="v$i load profile", color=Cycled(i))
+    end
+    axislegend(ax, position=:rc)
+    _fig
+end
Example block output

Lastly we want to observe two internal states of the pipes: the Reynolds number and the mean pressure. We see, that we're purely in the turbulent flow regime.

fig = begin
+    _fig = Figure()
+    row = 1
+    ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="Reynolds number", title="Reynolds number", xticks)
+    xlims!(ax, sol.t[begin], sol.t[end])
+    for i in 1:3
+        lines!(ax, sol, idxs=eidxs(nw, i, :Re); label="e $i", color=Cycled(i))
+    end
+    hlines!(ax, 2300, color=:black, linestyle=:dash, label="L/T transition")
+    axislegend(ax, position=:rb)
+    row += 1
+
+    ax = Axis(_fig[row, 1]; xlabel="time [h]", ylabel="Mean pressure [Pa]", title="Mean pressure in pipes", xticks)
+    xlims!(ax, sol.t[begin], sol.t[end])
+    for i in 1:3
+        lines!(ax, sol, idxs=eidxs(nw, i, :pM); label="e $i", color=Cycled(i))
+    end
+    axislegend(ax, position=:rb)
+    _fig
+end
Example block output

This page was generated using Literate.jl.

diff --git a/previews/PR184/generated/getting_started_with_network_dynamics.jl b/previews/PR184/generated/getting_started_with_network_dynamics.jl new file mode 100644 index 00000000..0638fd69 --- /dev/null +++ b/previews/PR184/generated/getting_started_with_network_dynamics.jl @@ -0,0 +1,157 @@ +# # Network Diffusion +# +# This introductory example explains the use of the basic types and constructors +# in NetworkDynamics.jl by modeling a simple diffusion on an undirected network. +# +# +# ## Theoretical background +# +# Diffusion processes are relevant for phenomena as diverse as heat conduction, electrical currents, and random walks. Generally speaking they describe the tendency of systems to evolve towards a state of equally distributed heat, charge or concentration. In such system the local temperature (or concentration) changes according to its difference with its neighborhood, i.e. the temperature gradient. +# +# Let $g$ be a graph with $N$ nodes and adjacency matrix $A$. Let $v = (v_1, \dots, v_n)$ be a vector of (abstract) temperatures or concentrations at each node $i = 1, \dots, N$. Then the rate of change of state $v_i$ is described by its difference with its neighbors and we obtain the following ordinary differential equation +# +# ```math +# \dot v_i = \sum_{j=1}^N A_{ji} (v_j - v_i). +# ``` +# +# The sum on the right hand side plays the role of a (discrete) gradient. If the temperature at node $i$ is higher than at its neighboring node $j$ it will decrease along that edge. +# +# ## Modeling diffusion in NetworkDynamics.jl +# We begin by loading the necessary packages. + +using Graphs +using NetworkDynamics +using OrdinaryDiffEqTsit5 +using StableRNGs +using Plots +nothing #hide + +# From the above considerations we see that in this model the nodes do not have any internal dynamics - if a node was disconnected from the rest of the network its state would never change, since then $A_{ji} = 0 \; \forall j$ and hence $\dot v_i = 0$. This means that the evolution of a node depends only on the interaction with its neighbors. In NetworkDynamics.jl, interactions with neighbors are described by equations for the edges. +# +# In order to bring this equation into the form required by NetworkDynamics.jl we need split the dynamics into edge and vertex parts and bring them into the correct input-output formulation. +# The vertices have one internal state $v$ which is also the output. The input is +# the sum over all flows of connected edges. This directly correspons to the component model definition outlined in Mathematical Model: +# ```math +# \begin{aligned} +# \dot x^\mathrm{v} &= f^{\mathrm v}(u^{\mathrm v}, \sum_k^{\text{incident}} y^{\mathrm e}_k, p^{\mathrm v}, t) &&= \sum_k^\mathrm{incident} y^\mathrm{e}_k \\ +# y^\mathrm{v} &= g^{\mathrm v}(u^{\mathrm v}, \sum_k^{\text{incident}} y^{\mathrm e}_k, p^{\mathrm v}, t) &&= x^\mathrm{v} +# \end{aligned} +# ``` +# The edge dynamics on the other hand do not have any internal states. Thus we +# only define the output as the difference between the source and destination +# vertex: +# ```math +# \begin{aligned} +# y^{\mathrm e}_{\mathrm{dst}} &= g_\mathrm{dst}^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t) &&= y^{\mathrm v}_{\mathrm{src}} - y^{\mathrm v}_{\mathrm{dst}}\\ +# y^{\mathrm e}_{\mathrm{src}} &= g_\mathrm{src}^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t) &&= y^{\mathrm v}_{\mathrm{dst}} - y^{\mathrm v}_{\mathrm{src}} +# \end{aligned} +# ``` +# +# ### Definition of `EdgeModel` + +function diffusionedge_g!(e_dst, v_src, v_dst, p, t) + # e_dst, v_src, v_dst are arrays, hence we use the broadcasting operator + e_dst .= v_src .- v_dst + nothing +end + +# The function `diffusionedge_g!` takes as inputs the current state of the edge `e`, its source vertex `v_src`, its destination vertex `v_dst`, a vector of parameters `p` and the time `t`. In order to comply with the syntax of NetworkDynamics.jl we always have to define functions for edges with exactly these arguments, even though we do not need `p` and `t` for the diffusion example. +# +# `diffusionedge_g!` is called a **mutating** function, since it modifies (or *mutates*) one of its inputs, namely the edge state `e`. As a convention in Julia names of mutating functions end with an `!`. The use of mutating functions reduces allocations and thereby speeds up computations. After the function call the edge's output value `e` equals the difference between its source and its destination vertex (i.e. the discrete gradient along that edge). +# +# Notably, this function only models $g_\mathrm{dst}$. However we can wrap this single-sided output function in an `AntiSymmetric` output wrapper to construct the `EdgeModel`: + +nd_diffusion_edge = EdgeModel(; g=AntiSymmetric(diffusionedge_g!), outsym=[:flow]) + +# ### Definition of `VertexModel` +# For undirected graphs, the `edgefunction!` specifies the coupling from a source- to a destination vertex. The contributions of the connected edges to a single vertex are "aggregated". Default aggregation is the summation of all incident edge states. The aggregated edge state is made available via the `esum` argument of the vertex function. + +function diffusionvertex_f!(dv, v, esum, p, t) + # dv, v and esum are arrays, hence we use the broadcasting operator . + dv .= esum + nothing +end + +# Just like above the input arguments `v, esum, p, t` are mandatory for the syntax of vertex functions. The additional input `dv` corresponding to the derivative of the vertex' state is mandatory for vertices described by ordinary differential equations. +# +# The output function `g` is just taking part of the internal states. For that we can use the `StateMask` helper function `g = StateMaks(1:1)` + +nd_diffusion_vertex = VertexModel(; f=diffusionvertex_f!, g=StateMask(1:1), dim=1) + +# ## Constructing the network +# +# With the components defined, we can define the topology and assemble the network dynamics. + +N = 20 # number of nodes +k = 4 # average degree +g = barabasi_albert(N, k) # a little more exciting than a bare random graph + +# The [Barabási–Albert model](https://en.wikipedia.org/wiki/Barab%C3%A1si%E2%80%93Albert_model) generates a scale-free random graph. + +nd = Network(g, nd_diffusion_vertex, nd_diffusion_edge) + +# The constructor `Network` combines the component model with the topological information contained in the graph **`g`** and returns an `Network` compatible with the solvers of `DifferentialEquations.jl`. + +rng = StableRNG(1) +x0 = randn(rng, N) # random initial conditions +ode_prob = ODEProblem(nd, x0, (0.0, 2.0)) +sol = solve(ode_prob, Tsit5()); + +# We are solving the diffusion problem on the time interval $[0, 2]$ with the `Tsit5()` algorithm, which is recommended by the authors of `DifferentialEquations.jl` for most non-stiff problems. + +plot(sol; idxs=vidxs(nd, :, :), fmt=:png) + +# The plotting is straightforward. The **`idxs`** keyword allows us to pass a list of indices. Indices can be also "symbolic" indices which specify components and their symbols directly. For example `idxs = VIndex(1, :v)` acesses state `:v` of vertex 1. See Symbolic Indexing for more details. +# +# In oder to collect multiple indices we can use the helper function `vidxs` and `eidxs`, which help to collect all symbolic indices matching a certain criteria. +# +# ## Two Dimensional Extension +# +# To illustrate a very simple multi-dimensional case, in the following we simulate two independent diffusions on an identical graph. The first uses the symbol `x` and is started with initial conditions drawn from the standard normal distribution $N(0,1)$, the second uses the symbol `ϕ` with squared standard normal inital conditions. +# +# The symbols have to be passed with the keyword **`sym`** to `VertexModel`. + +N = 10 # number of nodes +k = 4 # average degree +g = barabasi_albert(N, k) # a little more exciting than a bare random graph + +# We will have two independent diffusions on the network, hence dim = 2 +nd_diffusion_vertex_2 = VertexModel(; f=diffusionvertex_f!, g=1:2, dim=2, sym=[:x, :ϕ]) +nd_diffusion_edge_2 = EdgeModel(; g=AntiSymmetric(diffusionedge_g!), outsym=[:flow_x, :flow_ϕ]) +nd_2 = Network(g, nd_diffusion_vertex_2, nd_diffusion_edge_2) + +x0_2 = vec(transpose([randn(rng, N) .^ 2 randn(rng, N)])) # x ~ N(0,1)^2; ϕ ~ N(0,1) +ode_prob_2 = ODEProblem(nd_2, x0_2, (0.0, 3.0)) +sol_2 = solve(ode_prob_2, Tsit5()); + +# Try plotting the variables ϕ_i yourself. [To write ϕ type \phi and press TAB] + +plot(sol_2; idxs=vidxs(nd_2, :, :x), fmt=:png) + +# Using the `eidxs` helper function we can also plot the flow variables + +plot(sol_2; idxs=eidxs(nd_2, :, :flow_x), fmt=:png) + +# ## Appendix: The network Laplacian $L$ +# +# The diffusion equation on a network can be rewritten as +# +# ```math +# \dot v_i = \sum_{j=1}^N A_{ji} v_j - d_i v_i = e_i^T A v - d_i v_i +# ``` +# +# where $d_i$ is the degree of node $i$ and $e_i^T$ is the $i$-th standard basis vector. Introducing the diagonal matrix $D$ that has the degree of node $i$ in its $i$-th row and the Laplacian matrix $L = D - A$ we arrive at +# +# ```math +# \dot v = e_i^T(A - D) v +# ``` +# +# and finally +# +# ```math +# \dot v = - L v +# ``` +# +# This is a linear system of ODEs and its solution is a matrix exponential. To study the asymptotic behaviour of the system it suffices to analyze the eigenspectrum of $L$. For this reason $L$ is an important construction in network science. + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/getting_started_with_network_dynamics/190ecac8.svg b/previews/PR184/generated/getting_started_with_network_dynamics/190ecac8.svg new file mode 100644 index 00000000..d81fc67f --- /dev/null +++ b/previews/PR184/generated/getting_started_with_network_dynamics/190ecac8.svg @@ -0,0 +1,88 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/getting_started_with_network_dynamics/6b3a5417.svg b/previews/PR184/generated/getting_started_with_network_dynamics/6b3a5417.svg new file mode 100644 index 00000000..43bd064b --- /dev/null +++ b/previews/PR184/generated/getting_started_with_network_dynamics/6b3a5417.svg @@ -0,0 +1,60 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/getting_started_with_network_dynamics/7d053527.svg b/previews/PR184/generated/getting_started_with_network_dynamics/7d053527.svg new file mode 100644 index 00000000..5ad60875 --- /dev/null +++ b/previews/PR184/generated/getting_started_with_network_dynamics/7d053527.svg @@ -0,0 +1,82 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/getting_started_with_network_dynamics/index.html b/previews/PR184/generated/getting_started_with_network_dynamics/index.html new file mode 100644 index 00000000..ffcff74a --- /dev/null +++ b/previews/PR184/generated/getting_started_with_network_dynamics/index.html @@ -0,0 +1,43 @@ + +Getting Started · NetworkDynamics
GitHub

Network Diffusion

This introductory example explains the use of the basic types and constructors in NetworkDynamics.jl by modeling a simple diffusion on an undirected network.

This example can be dowloaded as a normal Julia script here.

Theoretical background

Diffusion processes are relevant for phenomena as diverse as heat conduction, electrical currents, and random walks. Generally speaking they describe the tendency of systems to evolve towards a state of equally distributed heat, charge or concentration. In such system the local temperature (or concentration) changes according to its difference with its neighborhood, i.e. the temperature gradient.

Let $g$ be a graph with $N$ nodes and adjacency matrix $A$. Let $v = (v_1, \dots, v_n)$ be a vector of (abstract) temperatures or concentrations at each node $i = 1, \dots, N$. Then the rate of change of state $v_i$ is described by its difference with its neighbors and we obtain the following ordinary differential equation

\[\dot v_i = \sum_{j=1}^N A_{ji} (v_j - v_i).\]

The sum on the right hand side plays the role of a (discrete) gradient. If the temperature at node $i$ is higher than at its neighboring node $j$ it will decrease along that edge.

Modeling diffusion in NetworkDynamics.jl

We begin by loading the necessary packages.

using Graphs
+using NetworkDynamics
+using OrdinaryDiffEqTsit5
+using StableRNGs
+using Plots

From the above considerations we see that in this model the nodes do not have any internal dynamics - if a node was disconnected from the rest of the network its state would never change, since then $A_{ji} = 0 \; \forall j$ and hence $\dot v_i = 0$. This means that the evolution of a node depends only on the interaction with its neighbors. In NetworkDynamics.jl, interactions with neighbors are described by equations for the edges.

In order to bring this equation into the form required by NetworkDynamics.jl we need split the dynamics into edge and vertex parts and bring them into the correct input-output formulation. The vertices have one internal state $v$ which is also the output. The input is the sum over all flows of connected edges. This directly correspons to the component model definition outlined in Mathematical Model:

\[\begin{aligned} +\dot x^\mathrm{v} &= f^{\mathrm v}(u^{\mathrm v}, \sum_k^{\text{incident}} y^{\mathrm e}_k, p^{\mathrm v}, t) &&= \sum_k^\mathrm{incident} y^\mathrm{e}_k \\ +y^\mathrm{v} &= g^{\mathrm v}(u^{\mathrm v}, \sum_k^{\text{incident}} y^{\mathrm e}_k, p^{\mathrm v}, t) &&= x^\mathrm{v} +\end{aligned}\]

The edge dynamics on the other hand do not have any internal states. Thus we only define the output as the difference between the source and destination vertex:

\[\begin{aligned} +y^{\mathrm e}_{\mathrm{dst}} &= g_\mathrm{dst}^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t) &&= y^{\mathrm v}_{\mathrm{src}} - y^{\mathrm v}_{\mathrm{dst}}\\ +y^{\mathrm e}_{\mathrm{src}} &= g_\mathrm{src}^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t) &&= y^{\mathrm v}_{\mathrm{dst}} - y^{\mathrm v}_{\mathrm{src}} +\end{aligned}\]

Definition of EdgeModel

function diffusionedge_g!(e_dst, v_src, v_dst, p, t)
+    # e_dst, v_src, v_dst are arrays, hence we use the broadcasting operator
+    e_dst .= v_src .- v_dst
+    nothing
+end

The function diffusionedge_g! takes as inputs the current state of the edge e, its source vertex v_src, its destination vertex v_dst, a vector of parameters p and the time t. In order to comply with the syntax of NetworkDynamics.jl we always have to define functions for edges with exactly these arguments, even though we do not need p and t for the diffusion example.

diffusionedge_g! is called a mutating function, since it modifies (or mutates) one of its inputs, namely the edge state e. As a convention in Julia names of mutating functions end with an !. The use of mutating functions reduces allocations and thereby speeds up computations. After the function call the edge's output value e equals the difference between its source and its destination vertex (i.e. the discrete gradient along that edge).

Notably, this function only models $g_\mathrm{dst}$. However we can wrap this single-sided output function in an AntiSymmetric output wrapper to construct the EdgeModel:

nd_diffusion_edge = EdgeModel(; g=AntiSymmetric(diffusionedge_g!), outsym=[:flow])
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ └─ 1/1 outputs: src=[₋flow] dst=[flow]

Definition of VertexModel

For undirected graphs, the edgefunction! specifies the coupling from a source- to a destination vertex. The contributions of the connected edges to a single vertex are "aggregated". Default aggregation is the summation of all incident edge states. The aggregated edge state is made available via the esum argument of the vertex function.

function diffusionvertex_f!(dv, v, esum, p, t)
+    # dv, v and esum are arrays, hence we use the broadcasting operator .
+    dv .= esum
+    nothing
+end

Just like above the input arguments v, esum, p, t are mandatory for the syntax of vertex functions. The additional input dv corresponding to the derivative of the vertex' state is mandatory for vertices described by ordinary differential equations.

The output function g is just taking part of the internal states. For that we can use the StateMask helper function g = StateMaks(1:1)

nd_diffusion_vertex = VertexModel(; f=diffusionvertex_f!, g=StateMask(1:1), dim=1)
VertexModel :VertexM PureStateMap()
+ ├─ 1 state:  [s]
+ └─ 1 output: [s]

Constructing the network

With the components defined, we can define the topology and assemble the network dynamics.

N = 20 # number of nodes
+k = 4  # average degree
+g = barabasi_albert(N, k) # a little more exciting than a bare random graph

The Barabási–Albert model generates a scale-free random graph.

nd = Network(g, nd_diffusion_vertex, nd_diffusion_edge)
Network with 20 states and 0 parameters
+ ├─ 20 vertices (1 unique type)
+ └─ 64 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

The constructor Network combines the component model with the topological information contained in the graph g and returns an Network compatible with the solvers of DifferentialEquations.jl.

rng = StableRNG(1)
+x0 = randn(rng, N) # random initial conditions
+ode_prob = ODEProblem(nd, x0, (0.0, 2.0))
+sol = solve(ode_prob, Tsit5());

We are solving the diffusion problem on the time interval $[0, 2]$ with the Tsit5() algorithm, which is recommended by the authors of DifferentialEquations.jl for most non-stiff problems.

plot(sol; idxs=vidxs(nd, :, :), fmt=:png)
Example block output

The plotting is straightforward. The idxs keyword allows us to pass a list of indices. Indices can be also "symbolic" indices which specify components and their symbols directly. For example idxs = VIndex(1, :v) acesses state :v of vertex 1. See Symbolic Indexing for more details.

In oder to collect multiple indices we can use the helper function vidxs and eidxs, which help to collect all symbolic indices matching a certain criteria.

Two Dimensional Extension

To illustrate a very simple multi-dimensional case, in the following we simulate two independent diffusions on an identical graph. The first uses the symbol x and is started with initial conditions drawn from the standard normal distribution $N(0,1)$, the second uses the symbol ϕ with squared standard normal inital conditions.

The symbols have to be passed with the keyword sym to VertexModel.

N = 10 # number of nodes
+k = 4  # average degree
+g = barabasi_albert(N, k) # a little more exciting than a bare random graph
+
+# We will have two independent diffusions on the network, hence dim = 2
+nd_diffusion_vertex_2 = VertexModel(; f=diffusionvertex_f!, g=1:2, dim=2, sym=[:x, :ϕ])
+nd_diffusion_edge_2 = EdgeModel(; g=AntiSymmetric(diffusionedge_g!), outsym=[:flow_x, :flow_ϕ])
+nd_2 = Network(g, nd_diffusion_vertex_2, nd_diffusion_edge_2)
+
+x0_2 = vec(transpose([randn(rng, N) .^ 2 randn(rng, N)])) # x ~ N(0,1)^2; ϕ ~ N(0,1)
+ode_prob_2 = ODEProblem(nd_2, x0_2, (0.0, 3.0))
+sol_2 = solve(ode_prob_2, Tsit5());

Try plotting the variables ϕ_i yourself. [To write ϕ type \phi and press TAB]

plot(sol_2; idxs=vidxs(nd_2, :, :x), fmt=:png)
Example block output

Using the eidxs helper function we can also plot the flow variables

plot(sol_2; idxs=eidxs(nd_2, :, :flow_x), fmt=:png)
Example block output

Appendix: The network Laplacian $L$

The diffusion equation on a network can be rewritten as

\[\dot v_i = \sum_{j=1}^N A_{ji} v_j - d_i v_i = e_i^T A v - d_i v_i\]

where $d_i$ is the degree of node $i$ and $e_i^T$ is the $i$-th standard basis vector. Introducing the diagonal matrix $D$ that has the degree of node $i$ in its $i$-th row and the Laplacian matrix $L = D - A$ we arrive at

\[\dot v = e_i^T(A - D) v\]

and finally

\[\dot v = - L v\]

This is a linear system of ODEs and its solution is a matrix exponential. To study the asymptotic behaviour of the system it suffices to analyze the eigenspectrum of $L$. For this reason $L$ is an important construction in network science.

This page was generated using Literate.jl.

diff --git a/previews/PR184/generated/heterogeneous_system.jl b/previews/PR184/generated/heterogeneous_system.jl new file mode 100644 index 00000000..454e2a4b --- /dev/null +++ b/previews/PR184/generated/heterogeneous_system.jl @@ -0,0 +1,149 @@ +# # Modeling a heterogeneous system +# +# +# One of the main purposes of NetworkDynamics.jl is to facilitate +# modeling coupled systems with heterogenities. This means that +# components can differ in their parameters as well as in their dynamics. +# +# ## Heterogenous parameters +# +# We start by setting up a simple system of Kuramoto oscillators. + +using NetworkDynamics, OrdinaryDiffEqTsit5, Plots, Graphs + +N = 8 +g = watts_strogatz(N, 2, 0) # ring network + +function kuramoto_edge!(e, θ_s, θ_d, (K,), t) + e[1] = K * sin(θ_s[1] - θ_d[1]) + nothing +end +edge! = EdgeModel(g=AntiSymmetric(kuramoto_edge!), outdim=1, psym=[:K=>3]) + +function kuramoto_vertex!(dθ, θ, esum, (ω0,), t) + dθ[1] = ω0 + esum[1] + nothing +end +vertex! = VertexModel(f=kuramoto_vertex!, g=StateMask(1:1), sym=[:θ], psym=[:ω0], name=:kuramoto) + +nw = Network(g, vertex!, edge!) + +# To assign parameters, we can create a `NWParameter` object based on the `nw` definition. +# This parameter object will be pre-filled with the default parameters. + +p = NWParameter(nw) + +# To set the vertex parameters, we can use indexing of the `p.v` field: + +ω = collect(1:N) ./ N +ω .-= sum(ω) / N +p.v[:, :ω0] = ω + +# Here, the index pairing `:, :ω` is used to index state ω for all node indices. +# +# The parameter object contains information about the network structure. For the +# actual problem definition we need to throw away this wrapper and use the +# flat-vector representation of the parameters `pflat(p)`. Note that `pflat(p)` +# +# Similarily, we could use `NWState(nw)` to create an indexable wrapper of the +# initial state. However in this case we can also fill create the flat state +# array manually: + +x0 = collect(1:N) ./ N +x0 .-= sum(x0) ./ N +tspan = (0.0, 10.0) +prob = ODEProblem(nw, x0, tspan, pflat(p)) +sol = solve(prob, Tsit5()) +plot(sol; ylabel="θ", fmt=:png) + +# ## Heterogeneous dynamics +# +# Two paradigmatic modifications of the node model above are static nodes and nodes with +# inertia. +# A static node has no internal states and instead fixes the variable at a +# constant value. + +function static_g(out, u, p, t) + out[1] = p[1] + nothing +end +static! = VertexModel(g=static_g, outsym=[:θ], psym=[:θfix => ω[1]], name=:static) + +# But wait! NetworkDynamics classified this as `PureFeedForward`, because it cannot +# distinguish between the function signatures +# ``` +# g(out, u, p, t) # PureFeedForward +# g(out, ins, p, t) # NoFeedForward +# ``` +# and since `dim(u)=0` it wrongfully assumes that the latter is meant. +# We can overwrite the classification by passing the ff keyword: + +static! = VertexModel(g=static_g, outsym=[:θ], psym=[:θfix => ω[1]], ff=NoFeedForward(), name=:static) + +# A Kuramoto model with inertia consists of two internal variables leading to +# more complicated (and for many applications more realistic) local dynamics. + +function kuramoto_inertia!(dv, v, esum, (ω0,), t) + dv[1] = v[2] + dv[2] = ω0 - 1.0 * v[2] + esum[1] + nothing +end + +inertia! = VertexModel(f=kuramoto_inertia!, g=1:1, sym=[:θ, :ω], psym=[:ω0], name=:inertia) + +# Since now we model a system with heterogeneous node dynamics we can no longer +# straightforwardly pass a single VertexModel to the `Network` constructor but +# instead have to hand over an Array. + +vertex_array = VertexModel[vertex! for i in 1:N] +vertex_array[1] = static! +vertex_array[5] = inertia! # index should correspond to the node's index in the graph +nw_hetero! = Network(g, vertex_array, edge!) + +# Now we have to take a bit more care with defining initial conditions and parameters. +# +# First, we can generate a `NWState` object based on the `nw_hetero!` object which +# will be populated with the default values. + +state = NWState(nw_hetero!) + +# The node with inertia is two-dimensional, hence we need to specify two initial conditions. +# For the first dimension we keep the initial conditions from above and insert! another one into `x0` at +# the correct index. +# +# For the θ states we will use the same initial conditins as before: + +state.v[2:8,:θ] = x0[2:8] + +# We're still missing one initial condition: the second variable ω of the 5th vertex. + +state.v[5,:ω] = 5 + +# The `NWState` object also contains a parameter object accessible via `state.p`. +# The edge parameters are already filled with default values. +# The vertex parameters can be copied from our old parmeter object `p`. + +state.p.v[2:8, :ω0] = p.v[2:8, :ω0] + +# For the problem construction, we need to convert the nested stuctures to flat arrays using the `uflat` and `pflat` methods. + +prob_hetero = ODEProblem(nw_hetero!, uflat(state), tspan, pflat(state)) +sol_hetero = solve(prob_hetero, Tsit5()); +plot(sol_hetero) + +# For clarity we plot only the variables referring to the oscillator's angle θ and color +# them according to their type. + +colors = map(vertex_array) do vertexf + if vertexf.name == :kuramoto + colorant"lightseagreen" + elseif vertexf.name == :static + colorant"orange" + elseif vertexf.name == :inertia + colorant"darkred" + end +end + +plot(sol_hetero; ylabel="θ", idxs=vidxs(1:8,:θ), lc=colors', fmt=:png) + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/heterogeneous_system/022b6041.svg b/previews/PR184/generated/heterogeneous_system/022b6041.svg new file mode 100644 index 00000000..8fec107d --- /dev/null +++ b/previews/PR184/generated/heterogeneous_system/022b6041.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/heterogeneous_system/5a435ac8.svg b/previews/PR184/generated/heterogeneous_system/5a435ac8.svg new file mode 100644 index 00000000..ba521c93 --- /dev/null +++ b/previews/PR184/generated/heterogeneous_system/5a435ac8.svg @@ -0,0 +1,62 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/heterogeneous_system/68c01707.svg b/previews/PR184/generated/heterogeneous_system/68c01707.svg new file mode 100644 index 00000000..ce3af32e --- /dev/null +++ b/previews/PR184/generated/heterogeneous_system/68c01707.svg @@ -0,0 +1,66 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/generated/heterogeneous_system/index.html b/previews/PR184/generated/heterogeneous_system/index.html new file mode 100644 index 00000000..62afcaeb --- /dev/null +++ b/previews/PR184/generated/heterogeneous_system/index.html @@ -0,0 +1,95 @@ + +Heterogeneous Systems · NetworkDynamics
GitHub

Modeling a heterogeneous system

This example can be dowloaded as a normal Julia script here.

One of the main purposes of NetworkDynamics.jl is to facilitate modeling coupled systems with heterogenities. This means that components can differ in their parameters as well as in their dynamics.

Heterogenous parameters

We start by setting up a simple system of Kuramoto oscillators.

using NetworkDynamics, OrdinaryDiffEqTsit5, Plots, Graphs
+
+N = 8
+g = watts_strogatz(N, 2, 0) # ring network
+
+function kuramoto_edge!(e, θ_s, θ_d, (K,), t)
+    e[1] = K * sin(θ_s[1] - θ_d[1])
+    nothing
+end
+edge! = EdgeModel(g=AntiSymmetric(kuramoto_edge!), outdim=1, psym=[:K=>3])
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[₋o] dst=[o]
+ └─   1 param:   [K=3]
function kuramoto_vertex!(dθ, θ, esum, (ω0,), t)
+    dθ[1] = ω0 + esum[1]
+    nothing
+end
+vertex! = VertexModel(f=kuramoto_vertex!, g=StateMask(1:1), sym=[:θ], psym=[:ω0], name=:kuramoto)
VertexModel :kuramoto PureStateMap()
+ ├─ 1 state:  [θ]
+ ├─ 1 output: [θ]
+ └─ 1 param:  [ω0]
nw = Network(g, vertex!, edge!)
Network with 8 states and 16 parameters
+ ├─ 8 vertices (1 unique type)
+ └─ 8 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

To assign parameters, we can create a NWParameter object based on the nw definition. This parameter object will be pre-filled with the default parameters.

p = NWParameter(nw)
Parameter{Vector{Float64}} of Network (8 vertices, 8 edges)
+  ├─ VPIndex(1, :ω0) => NaN
+  ├─ VPIndex(2, :ω0) => NaN
+  ├─ VPIndex(3, :ω0) => NaN
+  ├─ VPIndex(4, :ω0) => NaN
+  ├─ VPIndex(5, :ω0) => NaN
+  ├─ VPIndex(6, :ω0) => NaN
+  ├─ VPIndex(7, :ω0) => NaN
+  ├─ VPIndex(8, :ω0) => NaN
+  ├─ EPIndex(1, :K)  => 3.0
+  ├─ EPIndex(2, :K)  => 3.0
+  ├─ EPIndex(3, :K)  => 3.0
+  ├─ EPIndex(4, :K)  => 3.0
+  ├─ EPIndex(5, :K)  => 3.0
+  ├─ EPIndex(6, :K)  => 3.0
+  ├─ EPIndex(7, :K)  => 3.0
+  └─ EPIndex(8, :K)  => 3.0

To set the vertex parameters, we can use indexing of the p.v field:

ω = collect(1:N) ./ N
+ω .-= sum(ω) / N
+p.v[:, :ω0] = ω

Here, the index pairing :, :ω is used to index state ω for all node indices.

The parameter object contains information about the network structure. For the actual problem definition we need to throw away this wrapper and use the flat-vector representation of the parameters pflat(p). Note that pflat(p)

Similarily, we could use NWState(nw) to create an indexable wrapper of the initial state. However in this case we can also fill create the flat state array manually:

x0 = collect(1:N) ./ N
+x0 .-= sum(x0) ./ N
+tspan = (0.0, 10.0)
+prob = ODEProblem(nw, x0, tspan, pflat(p))
+sol = solve(prob, Tsit5())
+plot(sol; ylabel="θ", fmt=:png)
Example block output

Heterogeneous dynamics

Two paradigmatic modifications of the node model above are static nodes and nodes with inertia. A static node has no internal states and instead fixes the variable at a constant value.

function static_g(out, u, p, t)
+    out[1] = p[1]
+    nothing
+end
+static! = VertexModel(g=static_g, outsym=[:θ], psym=[:θfix => ω[1]], name=:static)
VertexModel :static PureFeedForward()
+ ├─ 0 states: []
+ ├─ 1 output: [θ]
+ └─ 1 param:  [θfix=-0.4375]

But wait! NetworkDynamics classified this as PureFeedForward, because it cannot distinguish between the function signatures

g(out, u, p, t)    # PureFeedForward
+g(out, ins, p, t)  # NoFeedForward

and since dim(u)=0 it wrongfully assumes that the latter is meant. We can overwrite the classification by passing the ff keyword:

static! = VertexModel(g=static_g, outsym=[:θ], psym=[:θfix => ω[1]], ff=NoFeedForward(), name=:static)
VertexModel :static NoFeedForward()
+ ├─ 0 states: []
+ ├─ 1 output: [θ]
+ └─ 1 param:  [θfix=-0.4375]

A Kuramoto model with inertia consists of two internal variables leading to more complicated (and for many applications more realistic) local dynamics.

function kuramoto_inertia!(dv, v, esum, (ω0,), t)
+    dv[1] = v[2]
+    dv[2] = ω0 - 1.0 * v[2] + esum[1]
+    nothing
+end
+
+inertia! = VertexModel(f=kuramoto_inertia!, g=1:1, sym=[:θ, :ω], psym=[:ω0], name=:inertia)
VertexModel :inertia PureStateMap()
+ ├─ 2 states: [θ, ω]
+ ├─ 1 output: [θ]
+ └─ 1 param:  [ω0]

Since now we model a system with heterogeneous node dynamics we can no longer straightforwardly pass a single VertexModel to the Network constructor but instead have to hand over an Array.

vertex_array    = VertexModel[vertex! for i in 1:N]
+vertex_array[1] = static!
+vertex_array[5] = inertia! # index should correspond to the node's index in the graph
+nw_hetero! = Network(g, vertex_array, edge!)
Network with 8 states and 16 parameters
+ ├─ 8 vertices (3 unique types)
+ └─ 8 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

Now we have to take a bit more care with defining initial conditions and parameters.

First, we can generate a NWState object based on the nw_hetero! object which will be populated with the default values.

state = NWState(nw_hetero!)
NWState{Vector{Float64}} of Network (8 vertices, 8 edges)
+  ├─ VIndex(2, :θ) => NaN
+  ├─ VIndex(3, :θ) => NaN
+  ├─ VIndex(4, :θ) => NaN
+  ├─ VIndex(6, :θ) => NaN
+  ├─ VIndex(7, :θ) => NaN
+  ├─ VIndex(8, :θ) => NaN
+  ├─ VIndex(5, :θ) => NaN
+  └─ VIndex(5, :ω) => NaN
+ p = NWParameter([-0.4375, NaN, NaN, NaN, NaN, NaN, NaN, NaN, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0])
+ t = nothing

The node with inertia is two-dimensional, hence we need to specify two initial conditions. For the first dimension we keep the initial conditions from above and insert! another one into x0 at the correct index.

For the θ states we will use the same initial conditins as before:

state.v[2:8,:θ] = x0[2:8]

We're still missing one initial condition: the second variable ω of the 5th vertex.

state.v[5,:ω] = 5

The NWState object also contains a parameter object accessible via state.p. The edge parameters are already filled with default values. The vertex parameters can be copied from our old parmeter object p.

state.p.v[2:8, :ω0] = p.v[2:8, :ω0]

For the problem construction, we need to convert the nested stuctures to flat arrays using the uflat and pflat methods.

prob_hetero = ODEProblem(nw_hetero!, uflat(state), tspan, pflat(state))
+sol_hetero = solve(prob_hetero, Tsit5());
+plot(sol_hetero)
Example block output

For clarity we plot only the variables referring to the oscillator's angle θ and color them according to their type.

colors = map(vertex_array) do vertexf
+    if vertexf.name == :kuramoto
+        colorant"lightseagreen"
+    elseif vertexf.name == :static
+        colorant"orange"
+    elseif vertexf.name == :inertia
+        colorant"darkred"
+    end
+end
+
+plot(sol_hetero; ylabel="θ", idxs=vidxs(1:8,:θ), lc=colors', fmt=:png)
Example block output

This page was generated using Literate.jl.

diff --git a/previews/PR184/generated/stress_on_truss.jl b/previews/PR184/generated/stress_on_truss.jl new file mode 100644 index 00000000..16428a87 --- /dev/null +++ b/previews/PR184/generated/stress_on_truss.jl @@ -0,0 +1,199 @@ +# # Stress on Truss +# In this exampe we'll simulate the time evolution of a truss structure consisting +# of joints and stiff springs. +# +# ![truss animation](truss.mp4) +# +# The mathematical model is quite simple: the vertices are point masses with +# positions $x$ and $y$ and velocities $v_x$ and $v_y$. +# For simplicity, we add some damping to the nodal motion based on the velocity. +# ```math +# \begin{aligned} +# \dot{x} &= v_x\\ +# \dot{y} &= v_y\\ +# \dot{v}_x &= \frac{1}{M}\left(\sum F_x - γ\,v_x\right)\\ +# \dot{v}_y &= \frac{1}{M}\left(\sum F_y - γ\,v_y\right) - g +# \end{aligned} +# ``` +# The vertices cannot absorb any torque, so the beams only exert forces in the direction of the beam. +# ```math +# |F| = K\cdot(L - |Δd|) +# ``` +# where $L$ is the nominal lenth and $|Δd|$ is the actual length of the beam. +# +# We start by loading the necessary packages. + +using NetworkDynamics +using OrdinaryDiffEqTsit5 +using Graphs +using GraphMakie +using LinearAlgebra: norm +using Printf +using CairoMakie +CairoMakie.activate!() + +# ## Definition of the dynamical system +# +# We need 3 models: +# - a fixed vertex which cannot change its position, +# - a free vertex which can move, and +# - a beam which connects two vertices. + +function fixed_g(pos, x, p, t) + pos .= p +end +vertex_fix = VertexModel(g=fixed_g, psym=[:xfix, :yfix], outsym=[:x, :y], ff=NoFeedForward()) + +# Here we need to specify the `ff` keyword manually, because NetworkDynamics cannot distinguish between +# `g(out, x, p, t)` (`NoFeedForwarwd`) and `g(out, in, p, t)` (`PureFeedForward()`) +# and guesses the latter when $\mathrm{dim}(x)=0$. + +function free_f(dx, x, Fsum, (M, γ, g), t) + v = view(x, 1:2) + dx[1:2] .= (Fsum .- γ .* v) ./ M + dx[2] -= g + dx[3:4] .= v + nothing +end +vertex_free = VertexModel(f=free_f, g=3:4, sym=[:vx=>0, :vy=>0, :x, :y], + psym=[:M=>10, :γ=>200, :g=>9.81], insym=[:Fx, :Fy]) + +# For the edge, we want to do something special. Later on, we want to color the edges according to the force they exert. +# Therefore, are interested in the absolut force rather than just the force vector. +# NetworkDynamics allows you to define so called `Observed` functions, which can recover additional states, +# so called observed, after the simulations. We can use this mechanis, to define a "recipe" for calculating the beam force +# based on the inputs (nodal positions) and the beam parameters. + +function edge_g!(F, pos_src, pos_dst, (K, L), t) + dx = pos_dst[1] - pos_src[1] + dy = pos_dst[2] - pos_src[2] + d = sqrt(dx^2 + dy^2) + Fabs = K * (L - d) + F[1] = Fabs * dx / d + F[2] = Fabs * dy / d + nothing +end +function observedf(obsout, u, pos_src, pos_dst, (K, L), t) + dx = pos_dst[1] .- pos_src[1] + dy = pos_dst[2] .- pos_src[2] + d = sqrt(dx^2 + dy^2) + obsout[1] = K * (L - d) + nothing +end +beam = EdgeModel(g=AntiSymmetric(edge_g!), psym=[:K=>0.5e6, :L], outsym=[:Fx, :Fy], obsf=observedf, obssym=[:Fabs]) + +# With the models define we can set up graph topology and initial positions. + +N = 5 +dx = 1.0 +shift = 0.2 +g = SimpleGraph(2*N + 1) +for i in 1:N + add_edge!(g, i, i+N); add_edge!(g, i, i+N) + if i < N + add_edge!(g, i+1, i+N); add_edge!(g, i, i+1); add_edge!(g, i+N, i+N+1) + end +end +add_edge!(g, 2N, 2N+1) +pos0 = zeros(Point2f, 2N + 1) +pos0[1:N] = [Point((i-1)dx,0) for i in 1:N] +pos0[N+1:2*N] = [Point(i*dx + shift, 1) for i in 1:N] +pos0[2N+1] = Point(N*dx + 1, -1) +fixed = [1,4] # set fixed vertices +nothing #hide + +# Now can collect the vertex models and construct the Network object. + +verts = VertexModel[vertex_free for i in 1:nv(g)] +for i in fixed + verts[i] = vertex_fix # use the fixed vertex for the fixed points +end +nw = Network(g, verts, beam) + +# In order to simulate the system we need to initialize the state and parameter vectors. +# Some states and parameters are shared between all vertices/edges. Those have been allready set +# in their constructors. The free symbols are +# - `x` and `y` for the position of the free vertices, +# - `xfix` and `yfix` for the position of the fixed vertices, +# - `L` for the nominal length of the beams. + +s = NWState(nw) +# set x/y and xfix/yfix +for i in eachindex(pos0, verts) + if i in fixed + s.p.v[i, :xfix] = pos0[i][1] + s.p.v[i, :yfix] = pos0[i][2] + else + s.v[i, :x] = pos0[i][1] + s.v[i, :y] = pos0[i][2] + end +end +# set L for edges +for (i,e) in enumerate(edges(g)) + s.p.e[i, :L] = norm(pos0[src(e)] - pos0[dst(e)]) +end +nothing #hide + +# Lastly there is a special vertex at the end of the truss which has a higher mass and reduced damping. + +s.p.v[11, :M] = 200 +s.p.v[11, :γ] = 100 +nothing #hide + +# No we have everything ready to build the ODEProblem and simulate the system. + +tspan = (0.0, 12.0) +prob = ODEProblem(nw, uflat(s), tspan, pflat(s)) +sol = solve(prob, Tsit5()) +nothing #hide + +# ## Plot the solution +# +# Plotting trajectories of points is kinda boring. So instead we're going to use +# [`GraphMakie.jl`](https://github.com/MakieOrg/GraphMakie.jl) to create a animation of the timeseries. + +fig = Figure(size=(1000,550)); +fig[1,1] = title = Label(fig, "Stress on truss", fontsize=30) +title.tellwidth = false + +fig[2,1] = ax = Axis(fig) +ax.aspect = DataAspect(); +hidespines!(ax); # no borders +hidedecorations!(ax); # no grid, axis, ... +limits!(ax, -0.1, pos0[end][1]+0.3, pos0[end][2]-0.5, 1.15) # axis limits to show full plot + +# get the maximum force during the simulation to get the color scale +# It is only possible to access `:Fabs` directly becaus we've define the observable function for it! +(fmin, fmax) = 0.3 .* extrema(Iterators.flatten(sol(sol.t, idxs=eidxs(nw, :, :Fabs)))) +p = graphplot!(ax, g; + edge_width = 4.0, + node_size = 3*sqrt.(try s.p.v[i, :M] catch; 10.0 end for i in 1:nv(g)), + nlabels = [i in fixed ? "Δ" : "" for i in 1:nv(g)], + nlabels_align = (:center,:top), + nlabels_fontsize = 30, + elabels = ["edge $i" for i in 1:ne(g)], + elabels_side = Dict(ne(g) => :right), + edge_color = [0.0 for i in 1:ne(g)], + edge_attr = (colorrange=(fmin,fmax), + colormap=:diverging_bkr_55_10_c35_n256)) + +# draw colorbar +fig[3,1] = cb = Colorbar(fig, get_edge_plot(p), label = "Axial force", vertical=false) + +T = tspan[2] +fps = 30 +trange = range(0.0, sol.t[end], length=Int(T * fps)) +record(fig, "truss.mp4", trange; framerate=fps) do t + title.text = @sprintf "Stress on truss (t = %.2f )" t + s_at_t = NWState(sol, t) + for i in eachindex(pos0) + p[:node_pos][][i] = (s_at_t.v[i, :x], s_at_t.v[i, :y]) + end + notify(p[:node_pos]) + load = s_at_t.e[:, :Fabs] + p.edge_color[] = load + p.elabels = [@sprintf("%.0f", l) for l in load] + fig +end + +# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl diff --git a/previews/PR184/generated/stress_on_truss/index.html b/previews/PR184/generated/stress_on_truss/index.html new file mode 100644 index 00000000..b987028f --- /dev/null +++ b/previews/PR184/generated/stress_on_truss/index.html @@ -0,0 +1,132 @@ + +Stress on Truss · NetworkDynamics
GitHub

Stress on Truss

In this exampe we'll simulate the time evolution of a truss structure consisting of joints and stiff springs. This example can be dowloaded as a normal Julia script here.

The mathematical model is quite simple: the vertices are point masses with positions $x$ and $y$ and velocities $v_x$ and $v_y$. For simplicity, we add some damping to the nodal motion based on the velocity.

\[\begin{aligned} +\dot{x} &= v_x\\ +\dot{y} &= v_y\\ +\dot{v}_x &= \frac{1}{M}\left(\sum F_x - γ\,v_x\right)\\ +\dot{v}_y &= \frac{1}{M}\left(\sum F_y - γ\,v_y\right) - g +\end{aligned}\]

The vertices cannot absorb any torque, so the beams only exert forces in the direction of the beam.

\[|F| = K\cdot(L - |Δd|)\]

where $L$ is the nominal lenth and $|Δd|$ is the actual length of the beam.

We start by loading the necessary packages.

using NetworkDynamics
+using OrdinaryDiffEqTsit5
+using Graphs
+using GraphMakie
+using LinearAlgebra: norm
+using Printf
+using CairoMakie
+CairoMakie.activate!()

Definition of the dynamical system

We need 3 models:

  • a fixed vertex which cannot change its position,
  • a free vertex which can move, and
  • a beam which connects two vertices.
function fixed_g(pos, x, p, t)
+    pos .= p
+end
+vertex_fix = VertexModel(g=fixed_g, psym=[:xfix, :yfix], outsym=[:x, :y], ff=NoFeedForward())
VertexModel :StaticVertexM NoFeedForward()
+ ├─ 0 states:  []
+ ├─ 2 outputs: [x, y]
+ └─ 2 params:  [xfix, yfix]

Here we need to specify the ff keyword manually, because NetworkDynamics cannot distinguish between g(out, x, p, t) (NoFeedForwarwd) and g(out, in, p, t) (PureFeedForward()) and guesses the latter when $\mathrm{dim}(x)=0$.

function free_f(dx, x, Fsum, (M, γ, g), t)
+    v = view(x, 1:2)
+    dx[1:2] .= (Fsum .- γ .* v) ./ M
+    dx[2] -= g
+    dx[3:4] .= v
+    nothing
+end
+vertex_free = VertexModel(f=free_f, g=3:4, sym=[:vx=>0, :vy=>0, :x, :y],
+                             psym=[:M=>10, :γ=>200, :g=>9.81], insym=[:Fx, :Fy])
VertexModel :VertexM PureStateMap()
+ ├─ 2 inputs:  [Fx, Fy]
+ ├─ 4 states:  [vx=0, vy=0, x, y]
+ ├─ 2 outputs: [x, y]
+ └─ 3 params:  [M=10, γ=200, g=9.81]

For the edge, we want to do something special. Later on, we want to color the edges according to the force they exert. Therefore, are interested in the absolut force rather than just the force vector. NetworkDynamics allows you to define so called Observed functions, which can recover additional states, so called observed, after the simulations. We can use this mechanis, to define a "recipe" for calculating the beam force based on the inputs (nodal positions) and the beam parameters.

function edge_g!(F, pos_src, pos_dst, (K, L), t)
+    dx = pos_dst[1] - pos_src[1]
+    dy = pos_dst[2] - pos_src[2]
+    d = sqrt(dx^2 + dy^2)
+    Fabs = K * (L - d)
+    F[1] = Fabs * dx / d
+    F[2] = Fabs * dy / d
+    nothing
+end
+function observedf(obsout, u, pos_src, pos_dst, (K, L), t)
+    dx = pos_dst[1] .- pos_src[1]
+    dy = pos_dst[2] .- pos_src[2]
+    d = sqrt(dx^2 + dy^2)
+    obsout[1] = K * (L - d)
+    nothing
+end
+beam = EdgeModel(g=AntiSymmetric(edge_g!), psym=[:K=>0.5e6, :L], outsym=[:Fx, :Fy], obsf=observedf, obssym=[:Fabs])
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ ├─ 2/2 outputs: src=[₋Fx, ₋Fy] dst=[Fx, Fy]
+ └─   2 params:  [K=5e5, L]

With the models define we can set up graph topology and initial positions.

N = 5
+dx = 1.0
+shift = 0.2
+g = SimpleGraph(2*N + 1)
+for i in 1:N
+    add_edge!(g, i, i+N); add_edge!(g, i, i+N)
+    if i < N
+        add_edge!(g, i+1, i+N); add_edge!(g, i, i+1); add_edge!(g, i+N, i+N+1)
+    end
+end
+add_edge!(g, 2N, 2N+1)
+pos0 = zeros(Point2f, 2N + 1)
+pos0[1:N] = [Point((i-1)dx,0) for i in 1:N]
+pos0[N+1:2*N] = [Point(i*dx + shift, 1) for i in 1:N]
+pos0[2N+1] = Point(N*dx + 1, -1)
+fixed = [1,4] # set fixed vertices

Now can collect the vertex models and construct the Network object.

verts = VertexModel[vertex_free for i in 1:nv(g)]
+for i in fixed
+    verts[i] = vertex_fix # use the fixed vertex for the fixed points
+end
+nw = Network(g, verts, beam)
Network with 36 states and 67 parameters
+ ├─ 11 vertices (2 unique types)
+ └─ 18 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

In order to simulate the system we need to initialize the state and parameter vectors. Some states and parameters are shared between all vertices/edges. Those have been allready set in their constructors. The free symbols are

  • x and y for the position of the free vertices,
  • xfix and yfix for the position of the fixed vertices,
  • L for the nominal length of the beams.
s = NWState(nw)
+# set x/y and xfix/yfix
+for i in eachindex(pos0, verts)
+    if i in fixed
+        s.p.v[i, :xfix] = pos0[i][1]
+        s.p.v[i, :yfix] = pos0[i][2]
+    else
+        s.v[i, :x] = pos0[i][1]
+        s.v[i, :y] = pos0[i][2]
+    end
+end
+# set L for edges
+for (i,e) in enumerate(edges(g))
+    s.p.e[i, :L] = norm(pos0[src(e)] - pos0[dst(e)])
+end

Lastly there is a special vertex at the end of the truss which has a higher mass and reduced damping.

s.p.v[11, :M] = 200
+s.p.v[11, :γ] = 100

No we have everything ready to build the ODEProblem and simulate the system.

tspan = (0.0, 12.0)
+prob = ODEProblem(nw, uflat(s), tspan, pflat(s))
+sol  = solve(prob, Tsit5())

Plot the solution

Plotting trajectories of points is kinda boring. So instead we're going to use GraphMakie.jl to create a animation of the timeseries.

fig = Figure(size=(1000,550));
+fig[1,1] = title = Label(fig, "Stress on truss", fontsize=30)
+title.tellwidth = false
+
+fig[2,1] = ax = Axis(fig)
+ax.aspect = DataAspect();
+hidespines!(ax); # no borders
+hidedecorations!(ax); # no grid, axis, ...
+limits!(ax, -0.1, pos0[end][1]+0.3, pos0[end][2]-0.5, 1.15) # axis limits to show full plot
+
+# get the maximum force during the simulation to get the color scale
+# It is only possible to access `:Fabs` directly becaus we've define the observable function for it!
+(fmin, fmax) = 0.3 .* extrema(Iterators.flatten(sol(sol.t, idxs=eidxs(nw, :, :Fabs))))
+p = graphplot!(ax, g;
+               edge_width = 4.0,
+               node_size = 3*sqrt.(try s.p.v[i, :M] catch; 10.0 end for i in 1:nv(g)),
+               nlabels = [i in fixed ? "Δ" : "" for i in 1:nv(g)],
+               nlabels_align = (:center,:top),
+               nlabels_fontsize = 30,
+               elabels = ["edge $i" for i in 1:ne(g)],
+               elabels_side = Dict(ne(g)  => :right),
+               edge_color = [0.0 for i in 1:ne(g)],
+               edge_attr = (colorrange=(fmin,fmax),
+                          colormap=:diverging_bkr_55_10_c35_n256))
+
+# draw colorbar
+fig[3,1] = cb = Colorbar(fig, get_edge_plot(p), label = "Axial force", vertical=false)
+
+T = tspan[2]
+fps = 30
+trange = range(0.0, sol.t[end], length=Int(T * fps))
+record(fig, "truss.mp4", trange; framerate=fps) do t
+    title.text = @sprintf "Stress on truss (t = %.2f )" t
+    s_at_t = NWState(sol, t)
+    for i in eachindex(pos0)
+        p[:node_pos][][i] = (s_at_t.v[i, :x], s_at_t.v[i, :y])
+    end
+    notify(p[:node_pos])
+    load = s_at_t.e[:, :Fabs]
+    p.edge_color[] = load
+    p.elabels = [@sprintf("%.0f", l) for l in load]
+    fig
+end
"truss.mp4"

This page was generated using Literate.jl.

diff --git a/previews/PR184/generated/truss.mp4 b/previews/PR184/generated/truss.mp4 new file mode 100644 index 00000000..42396010 Binary files /dev/null and b/previews/PR184/generated/truss.mp4 differ diff --git a/previews/PR184/index.html b/previews/PR184/index.html new file mode 100644 index 00000000..ad5df9b0 --- /dev/null +++ b/previews/PR184/index.html @@ -0,0 +1,527 @@ + +General · NetworkDynamics
GitHub

NetworkDynamics

A package for working with dynamical systems on complex networks. NetworkDynamics.jl provides an interface between Graphs.jl and DifferentialEquations.jl. It allows for high performant simulation of dynamic networks by describing the local dynamics on the edges and vertices of the graph.

The behavior of a node or an edge can be described by algebraic equations, by differential algebraic equation (DAEs) in mass matrix form or by ordinary differential equations (ODE).

The central construction is the function Network that receives functions describing the local dynamics on the edges and nodes of a graph g as inputs, and returns a composite function compatible with the DifferentialEquations.jl calling syntax.

nd = Network(g, vertex_dynamics,  edge_dynamics)
+nd(dx, x, p, t)

Main features:

  • Clear separation of local dynamics and topology: you can easily change topology of your system or switching out dynamical components.
  • High performance when working with heterogeneous models (which means heaving different local dynamics in different parts of your network).
  • Symbolic Indexing into solutions and states: NetworkDynamics keeps track of the states of the individual subsystems.
  • Different execution schemes: NetworkDynamics exploits the known inter-dependencies between components to auto parallelize execution, even on GPUs!
  • Equation based models: use ModelingToolkit.jl to define local dynamics, use NetworkDynamics.jl to combine them into large networks!

Where to begin?

Check out the Mathematical Model to understand the underlying modelling ideas of NetworkDynamics followed by the page on Network Construction to learn how to implement you own models.

If you prefer to look at some concrete code first check out the Getting Started tutorial!

Installation

Installation is straightforward with Julia's package manager.

(v1.11) pkg> add NetworkDynamics

Reproducibility

Direct dependencies used for this documentation:
Status `~/work/NetworkDynamics.jl/NetworkDynamics.jl/docs/Project.toml`
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+  [f489334b] StyledStrings v1.11.0
+Info Packages marked with ⌃ have new versions available and may be upgradable.
Julia Version:
Julia Version 1.11.1
+Commit 8f5b7ca12ad (2024-10-16 10:53 UTC)
+Build Info:
+  Official https://julialang.org/ release
+Platform Info:
+  OS: Linux (x86_64-linux-gnu)
+  CPU: 4 × AMD EPYC 7763 64-Core Processor
+  WORD_SIZE: 64
+  LLVM: libLLVM-16.0.6 (ORCJIT, znver3)
+Threads: 1 default, 0 interactive, 1 GC (on 4 virtual cores)
+Environment:
+  JULIA_PKG_SERVER_REGISTRY_PREFERENCE = eager
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+Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`

Funding

Development of this project was in part funded by the German Federal Ministry for Economic Affairs and Climate Action as part of the OpPoDyn-Project (Project ID 01258425/1, 2024-2027).

diff --git a/previews/PR184/initialization/index.html b/previews/PR184/initialization/index.html new file mode 100644 index 00000000..f3328c5e --- /dev/null +++ b/previews/PR184/initialization/index.html @@ -0,0 +1,40 @@ + +Initialization · NetworkDynamics
GitHub

Initialization

Initialization of the system describes the process of finding valid initial conditions, mostly a fixpoint of the system. We distinguish between two types of initialization: full system initialziation and component initialization.

Full-System Initialization

Full system initialization describs the process of finding a fixpoint/steady state of th entire system.

To do so, you can use find_fixpoint, which creates a SteadyStateProblem of the whole network and tries do solve it.

Component-wise Initialization

In contrast to full-system initialization the goal of component-wise initialization is to find a valid initial condition for a single component first, given a network coupling.

This can be usefull in cases, where there are nontrivial internal dynamics and states within a single vertex or edge. The idea of component-wise initialisation is to find internal states which match a given "network coupling" (fixed inputs and outputs).

Lets consider the following example of a Swing-equation generator model.

using NetworkDynamics, ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as Dt
+
+@mtkmodel Swing begin
+    @variables begin
+        u_r(t)=1, [description="bus d-voltage", output=true]
+        u_i(t)=0.1, [description="bus q-voltage", output=true]
+        i_r(t)=1, [description="bus d-current (flowing into bus)", input=true]
+        i_i(t)=0.1, [description="bus d-current (flowing into bus)", input=true]
+        ω(t), [guess=0.0, description="Rotor frequency"]
+        θ(t), [guess=0.0, description="Rotor angle"]
+        Pel(t), [guess=1, description="Electrical Power injected into the grid"]
+    end
+    @parameters begin
+        M=0.005, [description="Inertia"]
+        D=0.1, [description="Damping"]
+        V=sqrt(u_r^2 + u_i^2), [description="Voltage magnitude"]
+        ω_ref=0, [description="Reference frequency"]
+        Pm, [guess=0.1,description="Mechanical Power"]
+    end
+    @equations begin
+        Dt(θ) ~ ω - ω_ref
+        Dt(ω) ~ 1/M * (Pm - D*ω - Pel)
+        Pel ~ u_r*i_r + u_i*i_i
+        u_r ~ V*cos(θ)
+        u_i ~ V*sin(θ)
+    end
+end
+sys = Swing(name=:swing)
+vf = VertexModel(sys, [:i_r, :i_i], [:u_r, :u_i])
VertexModel :swing NoFeedForward()
+ ├─ 2 inputs:  [i_r=1, i_i=0.1]
+ ├─ 2 states:  [θ≈0, ω≈0]
+ ├─ 2 outputs: [u_r=1, u_i=0.1]
+ └─ 5 params:  [M=0.005, Pm≈0.1, V=1.005, D=0.1, ω_ref=0]

You can see in the provided metadata, that we've set default values for the node outputs u_r, u_i, the node inputs i_r, i_i and most parameters. For some states and parameters, we've onlye provided a guess rather than a default. Variables which only have guesses are considered "tunable" for the initialization algorithm.

In order to initialize the remaining variables we use initialize_component!, which is a mutating function which tries to solve the nonlinear initialization problem and store the found values for the "free" variables as init metadata.

initialize_component!(vf; verbose=true)
[ Info: Initialization problem is overconstrained (3 vars for 4 equations). Create NonlinearLeastSquaresProblem for [:θ, :ω, :Pm].
+[ Info: Initialization successful with residual 2.2204742383278642e-14
VertexModel :swing NoFeedForward()
+ ├─ 2 inputs:  [i_r=1, i_i=0.1]
+ ├─ 2 states:  [θ=0.099669, ω=3.4499e-23]
+ ├─ 2 outputs: [u_r=1, u_i=0.1]
+ └─ 5 params:  [M=0.005, Pm=1.01, V=1.005, D=0.1, ω_ref=0]

Which lead to a successfull initialization of states :θ and :ω as well as parameter :Pm. To retrieve the residual you can use init_residual.

As a quick test we can ensure that the angle indeed matches the voltag angel:

get_init(vf, :θ) ≈ atan(get_default(vf, :u_i), get_default(vf, :u_r))
true
diff --git a/previews/PR184/mathematical_model/index.html b/previews/PR184/mathematical_model/index.html new file mode 100644 index 00000000..9dff0d75 --- /dev/null +++ b/previews/PR184/mathematical_model/index.html @@ -0,0 +1,48 @@ + +Mathematical Model · NetworkDynamics
GitHub

Mathematical Model

The basic mathematical model of NetworkDynamics.jl splits up the system it two parts: vertex and edge components.

The main goal of NetworkDynamics.jl is, to express the overall network dynamics as a Differential-Algebraic-Equation (DAE)

\[M\,\frac{\mathrm{d}}{\mathrm{d}t}u = f^{\mathrm{nw}}(u, p, t)\]

where M is a (possibly singular) mass matrix, $u$ is the internal state vector of the system, $p$ are the parameters and $t$ is the time. To make this compatible with the solvers for OrdinaryDiffEq.jl, the created Network object is a callable object

nw(du, u, p, t) # mutates du

with stored mass matrix information to build an ODEProblem based on the Network.

Instead of defining $f^{\mathrm{nw}}$ by hand, ND.jl helps you to build it automatically based on a list of decentralized nodal and edge dynamics, so-called VertexModel and EdgeModel objects. Each component model $\mathrm c$ is modeled as general input-output-system

\[\begin{aligned} +M_{\mathrm c}\,\frac{\mathrm{d}}{\mathrm{d}t}x_{\mathrm c} &= f^{\mathrm c}(x^{\mathrm c}, i_{\mathrm c}, p_{\mathrm c}, t)\\ +y^{\mathrm c} &= g^{\mathrm c}(x^\mathrm{c}, i_{\mathrm c}, p_{\mathrm c}, t) +\end{aligned}\]

where $M_{\mathrm{c}}$ is the component mass matrix, $x^{\mathrm c}$ are the component states, $i^{\mathrm c}$ are the inputs of the component and $y^{\mathrm c}$ is the output of the component. It is possible to have $\mathrm{dim}(x^{\mathrm{c}}) = 0$ and thus no internal states.

In the network context, the output of the edges are flow variables. The outputs of vertices are potential variables. In interconnection, the flow on the edges depends on the potentials at both ends as inputs. The potentials of the nodes depend on the incoming flows from all connected edges as an input. (Here, flow and potentials are meant in a conceptional and not necessarily physical way.)

Vertex Models

Specifically, a (single-layer) vertex model has one input, and one output. The input is an aggregation/reduction over all incident edge outputs,

\[i^{\mathrm v} = \mathop{\mathrm{agg}}\limits_k^{\text{incident}} y^{\mathrm e}_k \qquad\text{often}\qquad +i^{\mathrm v} = \sum_k^{\text{incident}} y^{\mathrm e}_k\]

The full vertex model

\[\begin{aligned} +M^{\mathrm v}\,\frac{\mathrm{d}}{\mathrm{d}t}x^{\mathrm v} &= f^{\mathrm v}(u^{\mathrm v}, i^{\mathrm v}, p^{\mathrm v}, t)\\ +y^{\mathrm v} &= g^{\mathrm v}(u^{\mathrm v}, i^{\mathrm v}, p^{\mathrm v}, t) +\end{aligned}\]

corresponds to the Julia functions

function fᵥ(dxᵥ, xᵥ, e_aggr, pᵥ, t)
+    # mutate dxᵥ
+    nothing
+end
+function gᵥ(yᵥ, xᵥ, e_aggr, pᵥ, t)
+    # mutate yᵥ
+    nothing
+end
+vertf = VertexModel(; f=fᵥ, g=gᵥ, mass_matrix=Mᵥ, ...)

Edge Models

In contrast to vertex models, edge models in general have two inputs and two outputs, both for source and destination end of the edge. We commonly use src and dst to describe the source and destination end of an edge respectively.

On the directionality of edges

Mathematically, in a system defined on an undirected graph there is no difference between the edge $(1,2)$ and $(2,1)$, the edge has no direction. However, from an implementation point of view we always need to have some kind of ordering for function arguments, state order and so on. For undirected graphs, Graphs.jl chooses the direction of an edge v1->v2 such that v1 < v2.

The inputs of the edge are just the outputs of the two nodes at both ends. The output is split into two: the dst output goes to the input of the vertex at the destination end, the src output goes to the input of the vertex at the src end.

The full model of an edge

\[\begin{aligned} +M^{\mathrm e}\,\frac{\mathrm{d}}{\mathrm{d}t}x^{\mathrm e} &= f^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t)\\ +y^{\mathrm e}_{\mathrm{dst}} &= g_\mathrm{dst}^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t)\\ +y^{\mathrm e}_{\mathrm{src}} &= g_\mathrm{src}^{\mathrm e}(u^{\mathrm e}, y^{\mathrm v}_{\mathrm{src}}, y^{\mathrm v}_{\mathrm{dst}}, p^{\mathrm e}, t) +\end{aligned}\]

corresponds to the Julia functions

function fₑ(dxₑ, xₑ, v_src, v_dst, pₑ, t)
+    # mutate dxᵥ
+    nothing
+end
+function gₑ(y_src, y_dst, xᵥ, v_src, v_dst, pₑ, t)
+    # mutate y_src and y_dst
+    nothing
+end
+vertf = EdgeModel(; f=fₑ, g=gₑ, mass_matrix=Mₑ, ...)

The sign convention for both outputs of an edge must be identical, typically, a positive flow represents a flow into the connected vertex. This is important, because the vertex only receives the flows, it does not know whether the flow was produce by the source or destination end of an edge.

          y_src     y_dst 
+  V_src o───←─────────→───o V_dst
+

Single Sided Edge Outputs

Often, edge outputs will possess some symmetry which makes it more convenient to define "single sided" edge output functions

function g_single(y, xᵥ, v_src, v_dst, pₑ, t)
+    # mutate y
+    nothing
+end

There are multiple wrappers available to automaticially convert them into double-sided edge output functions:

  • Directed(g_single) builds a double-sided function which only couples to the destination side.
  • Symmetric(g_single) builds a double-sided function in which both ends receive y.
  • AntiSymmetric(g_single) builds a double-sided function where the destination receives y and the source receives -y.
  • Fiducial(g_single_src, g_singl_dst) builds a double-sided edge output function based on two single sided functions.

Feed Forward Behavior

The most general version of the component models can contain direct feed forwards from the input, i.e. the edge output might depend directly on the connected vertices or the vertex output might depend directly on the aggregated edge input.

Whenever possible, you should define output functions without feed forwards, i.e.

gᵥ_noff(yᵥ, xᵥ, pᵥ, t)
+gₑ_noff([y_src,] y_dst, xᵥ, pₑ, t)

instead of the more general

gᵥ(yᵥ, xᵥ, e_aggr, pᵥ, t)
+gₑ([y_src], y_dst, xᵥ, v_src, v_dst, pₑ, t)

NetworkDynamics cannot couple two components with feed forward to each other. It is always possible to transform feed forward behavior to an internal state x with mass matrix entry zero to circumvent this problem. This transformation can be performed automatically by using ff_to_constraint.

Feed Forward Vertices

As of 11/2024, vertices with feed forward are not supported at all. Use ff_to_constraint to transform them into vertex model without FF.

Concretely, NetworkDynamics distinguishes between 4 types of feed forward behaviors of g functions based on the FeedForwardType trait. The different types the signature of provided function g. Based on the signatures avaialable, ND.jl will try to find the correct type automaticially. Using the ff keyword in the constructors, the user can enforce a specific type.

PureFeedForward()

g!(outs...,          ins...,       p, t) # abstractly
+g!(out_dst,          v_src, v_dst, p, t) # single-sided edge
+g!(out_src, out_dst, v_src, v_dst, p, t) # double-sided edge
+g!(v_out,            e_aggr,       p, t) # single layer vertex

FeedForward()

g!(outs...,          x, ins...,       p, t) # abstractly
+g!(out_dst,          x, v_src, v_dst, p, t) # single-sided edge
+g!(out_src, out_dst, x, v_src, v_dst, p, t) # double-sided edge
+g!(v_out,            x, e_aggr,       p, t) # single layer vertex

NoFeedForward()

g!(outs...,          x, p, t) # abstractly
+g!(out_dst,          x, p, t) # single-sided edge
+g!(out_src, out_dst, x, p, t) # double-sided edge
+g!(v_out,            x, p, t) # single layer vertex

PureStateMap()

g!(outs...,          x) # abstractly
+g!(out_dst,          x) # single-sided edge
+g!(out_src, out_dst, x) # double-sided edge
+g!(v_out,            x) # single layer vertex
diff --git a/previews/PR184/metadata/index.html b/previews/PR184/metadata/index.html new file mode 100644 index 00000000..53bd327e --- /dev/null +++ b/previews/PR184/metadata/index.html @@ -0,0 +1,2 @@ + +Metadata · NetworkDynamics
GitHub

Metadata

Component model such as VertexModel and EdgeModel can store metadata. We distinguish between two kinds of metadata: component metadata and symbol metadata.

Component Metadata

Component metadata is a Dict{Symbol,Any} attached to each component to store various information. Use metadata to retrieve the full dict.

To access the data, you can use the methods has_metadata, get_metadata and set_metadata! (see Component Metadata API).

Special metadata:

  • :init_residual: after Component-wise Initialization, this field stores the residual vector of the nonlinear problem.
  • :graphelement: optional field to specialize the graphelement for each component (vidx) for vertices, (;src,dst) named tuple of either vertex names or vertex indices for edges. Has special accessors has_/get_/set_graphelement.

Symbol Metadata

Each component stores symbol metadata. The symbol metadata is a Dict{Symbol, Dict{Symbol, Any}} which stores a metadate dict per symbol. Symbols are everything that appears in sym, psym, obssym and insym.

To access the data, you can use the methods has_metadata, get_metadata and set_metadata! (see Per Symbol Metadata API).

Special cases for symbol metadata are:

  • default: Stores default values for states/parameters. In initialization, those are considered fixed.
  • guess: Stores a guess for a state/parameter which needs to solved during initialization ("free" variables).
  • bounds: Store bounds for variables/parameters
  • init: Stores the solution of the "free" variables during initialization.

Fore those, there are special functions has_*, get_* and set_*!. See Per Symbol Metadata API.

Those are closely aligned to the metadata use in ModelingToolkit. They are automatically copied from the ODESystem if you use MTK models to create NetworkDynamics models.

diff --git a/previews/PR184/mtk_integration/6aeb76f8.png b/previews/PR184/mtk_integration/6aeb76f8.png new file mode 100644 index 00000000..194b97fc Binary files /dev/null and b/previews/PR184/mtk_integration/6aeb76f8.png differ diff --git a/previews/PR184/mtk_integration/index.html b/previews/PR184/mtk_integration/index.html new file mode 100644 index 00000000..384f03e5 --- /dev/null +++ b/previews/PR184/mtk_integration/index.html @@ -0,0 +1,84 @@ + +ModelingToolkit Integration · NetworkDynamics
GitHub

ModelingToolkit Integration

NetworkDynamics.jl is compatible with ModelingTookit.jl (MTK). The general idea is to use MTK to define component models (i.e. edge and vertex dynamics) which are then connected on network scale using NetworkDynamics.

The main entry point for this interop are the constructors

VertexModel(::ODESystem, inputs, outputs)
+EdgeModel(::ODESystem, srcin, dstin, [srscout], dstout)

whose docstrings can be found in the Component Models with MTK section in the API.

These constructors will:

  • transforming the states marked as input to parameters and structural_simplifying the system,
  • generating the f and g functions,
  • generate code for observables,
  • port all supported Metadata from MTK symbols to component symbols and
  • output a Vertex-/EdgeModel function compatible with NetworkDynamics.jl.

The main usecase for this feature is when you want to build relatively complex component models but interconnect them in a very homogeneous way (i.e. having the same output/input pairings in the whole system).

In theory, you can achieve everything you want to do with plain MTK. The idea of combining the two is, that NetworkDynamics offers far less flexibility when in comes to interconnection of subsystems on the network level. This might allow ND to exploit more knowledge of the structure without very expensive operations such as tearing of thousands of equations.

Warning

ModelingToolkit is a fast paced library with lots of functionality and ever growing complexity. As such the provided interface is kinda experimental. Some features of MTK are straight up unsupported, for example events within models or delay differential equations.

RC-Circuit Example

In good MTK tradition, this feature will be explained along a simple RC circuit example. The Dynamic Flow in simple Gas Network example is another showcase of the MTK constructors.

The system to model is 2 node, 1 edge network. The node output states are the voltage (to ground), the edge output sates are the currents at both ends.


+ideal v source      Resistor     Capacitor
+            v1 o─←────MMM────→─o v2
+               │               ┴ 
+              (↗)              ┬
+               │               │
+               ⏚               ⏚

Obviously there is no need in modeling such a small system using NetworkDynamics, however the method extends quite easily to construct large electrical networks reusing the same fundamental building blocks.

using NetworkDynamics
+using ModelingToolkit
+using ModelingToolkit: t_nounits as t, D_nounits as D
+using OrdinaryDiffEqTsit5
+using CairoMakie

All our components have "terminals", which have a voltage and current. We don't use the @connector from MTK here because our pins mark the interface towards the network and do not follow the MTK connector semantics.

@mtkmodel NWTerminal begin
+    @variables begin
+        v(t), [description="Voltage at node"]
+        i(t), [description="Current flowing into node"]
+    end
+end

An ideal voltage source is just a model which pins its output voltage to a fixed parameter. The source ejects whatever current is necessary. We introduce another variable i(t) to "capture" this current. This variable will be removed during structural simplify, but will be available for plotting through the Observables mechanism. The VertexModel can be generated from an ODESystem by providing names of the input and output states:

@mtkmodel VoltageSource begin
+    @components begin
+       p = NWTerminal()
+    end
+    @parameters begin
+        V = 1.0
+    end
+    @variables begin
+        i(t), [description="produced current by ideal voltage source (observable)"]
+    end
+    @equations begin
+        i ~ -p.i
+        p.v ~ V
+    end
+end
+@named vs = VoltageSource()
+vs_vertex = VertexModel(vs, [:p₊i], [:p₊v]; vidx=1)
VertexModel :vs NoFeedForward() @ Vertex 1
+ ├─ 1 input:  [p₊i]
+ ├─ 0 states: []
+ ├─ 1 output: [p₊v]
+ └─ 1 param:  [V=1]

A capacitor is a slightly more complicated model. Its voltage is defined as an differential equation based on the inflowing current.

@mtkmodel Capacitor begin
+    @components begin
+        p = NWTerminal(;v=0)
+    end
+    @parameters begin
+        C = 1.0
+    end
+    @equations begin
+        D(p.v) ~ p.i / C
+    end
+end
+@named cap = Capacitor()
+cap_vertex = VertexModel(cap, [:p₊i], [:p₊v], vidx=2)
VertexModel :cap PureStateMap() @ Vertex 2
+ ├─ 1 input:  [p₊i]
+ ├─ 1 state:  [p₊v=0]
+ ├─ 1 output: [p₊v=0]
+ └─ 1 param:  [C=1]

For the resistor we need two pins, one for the src and one for the dst side. The equations are straight forward.

@mtkmodel Resistor begin
+    @components begin
+        src = NWTerminal()
+        dst = NWTerminal()
+    end
+    @parameters begin
+        R = 1.0
+    end
+    @equations begin
+        dst.i ~ (src.v - dst.v)/R
+        src.i ~ -dst.i
+    end
+end
+@named resistor = Resistor()
+resistor_edge = EdgeModel(resistor, [:src₊v], [:dst₊v], [:src₊i], [:dst₊i]; src=:vs, dst=:cap)
EdgeModel :resistor PureFeedForward() @ Edge vs=>cap
+ ├─ 1/1 inputs:  src=[src₊v] dst=[dst₊v]
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[src₊i] dst=[dst₊i]
+ └─   1 param:   [R=1]

Having all those components defined, we can build the network. We don't need to provide a graph object here because we specified the placement in the graph on a per component basis.

nw = Network([vs_vertex, cap_vertex], [resistor_edge])
Network with 1 state and 3 parameters
+ ├─ 2 vertices (2 unique types)
+ └─ 1 edges (1 unique type)
+Edge-Aggregation using SequentialAggregator(+)

We can see, that NetworkDynamics internally is able to reduce all of the "output" states. We end up with a plain ODE of a single state.

Now we can simulate the system. For that we generate the u0 object. Since the metadata (such as default values) was automatically transferred, we can straight away construct the ODEProblem and solve the system.

u0 = NWState(nw) # generate state based on default values
+prob = ODEProblem(nw, uflat(u0), (0, 10.0), pflat(u0))
+sol = solve(prob, Tsit5())
+
+# plot the solution
+fig, ax1, p = plot(sol; idxs=VIndex(:cap, :p₊v), label="Capacitor Voltage");
+axislegend(ax1)
+ax2 = Axis(fig[2,1])
+plot!(ax2, sol; idxs=VIndex(:vs, :i), label="Current produced by ideal v source")
+axislegend(ax2)
Example block output
diff --git a/previews/PR184/network_construction/index.html b/previews/PR184/network_construction/index.html new file mode 100644 index 00000000..6ac728da --- /dev/null +++ b/previews/PR184/network_construction/index.html @@ -0,0 +1,61 @@ + +Network Construction · NetworkDynamics
GitHub

Network Construction

Building a Network

The main type of NetworkDyanmics.jl is a Network. A network bundles various component models (edge and vertex models) together with a graph to form a callable object which represents the RHS of the overall dynamical system, see Mathematical Model.

A Network is build by passing a graph g, vertex models vertexm and edge models edgem.

nw = Network(g, vertexm, edgem; kwargs...)

Two important keywords for the Network constructor are:

  • execution: Defines the ExecutionStyle of the coreloop, e.g. SequentialExecution{true}(). A execution style is a special struct which tells the backend how to parallelize for example. A list of available executions styles can be found under Execution Types in the API.

  • aggregator: Tells the backend how to aggregate and which aggregation function to use. Aggregation is the process of creating a single vertex input by reducing over the outputs of adjecent edges of said vertex. The aggregator contains both the function and the algorith. E.g. SequentialAggregator(+) is a sequential aggregation by summation. A list of availabe Aggregators can be found under Aggregators in the API.

Building VertexModels

This chapter walks through the most important aspects when defining custom vertex model. For a list of all keyword arguments please check out the docstring of VertexModel. As an example, we'll construct an second order kuramoto model, because that's what we do.

function kuramoto_f!(dv, v, esum, p, t)
+    M, P, D = p
+    dv[1] = v[2]
+    dv[2] = (P - D*v[2] + esum[1])/M
+    nothing
+end
+function kuramoto_g!(y, v, esum, p, t)
+    y[1] = v[1]
+    nothing
+end
+VertexModel(; f=kuramoto_f!, g=kuramoto_g!, dim=2, pdim=3, outdim=1)
VertexModel :VertexM FeedForward()
+ ├─ 2 states: [v₁, v₂]
+ ├─ 1 output: [o]
+ └─ 3 params: [p₁, p₂, p₃]

Those keywords are the minimum metadata we need to provide.

However there is a problem: the vertex is classified as a FeedForward vertex, which is unnecessary. We can improve the implementation of g according to the Feed Forward Behavior section.

function kuramoto_g_noff!(y, v, p, t)
+    y[1] = v[1]
+    nothing
+end
+VertexModel(; f=kuramoto_f!, g=kuramoto_g_noff!, dim=2, pdim=3, outdim=1)
VertexModel :VertexM NoFeedForward()
+ ├─ 2 states: [v₁, v₂]
+ ├─ 1 output: [o]
+ └─ 3 params: [p₁, p₂, p₃]

It is still annoying to explicitly write this trivial output function. You can prevent this by using StateMask. By writing

VertexModel(; f=kuramoto_f!, g=StateMask(1:1), dim=2, pdim=3)
VertexModel :VertexM PureStateMap()
+ ├─ 2 states: [v₁, v₂]
+ ├─ 1 output: [v₁]
+ └─ 3 params: [p₁, p₂, p₃]

we told the vertex model, that the output is part of the states x[1:1]. This enables a few things:

  • outdim is not needed anymore, can be inferred from StateMask
  • outsym is not a generic :o any more but inferred from the state symbols.

We can be even less verbose by writing g=1:1 or just g=1.

In a last we define better names for our states and parameters as well as assigning a position in the graph to enable the graphless network construction. Whenever you provide sym keyword the corresponding dim keyword is not neccessary anymore. We end up with a relatively short definition

VertexModel(; f=kuramoto_f!, g=1,
+              sym=[:θ, :ω], psym=[:M=>1, :P=>0.1, :D=>0],
+              insym=[:P_nw], name=:swing, vidx=1)
VertexModel :swing PureStateMap() @ Vertex 1
+ ├─ 1 input:  [P_nw]
+ ├─ 2 states: [θ, ω]
+ ├─ 1 output: [θ]
+ └─ 3 params: [M=1, P=0.1, D=0]

Building EdgeModels

This chapter walks through the most important aspects when defining custom edge models. For a list of all keyword arguments please check out the docstring of EdgeModel.

As an example edge model we want to define standard sinusoidal coupling between the vertices in our network. The full definition looks like this:

function edge_f!(de, e, vsrc, vdst, p, t)
+    nothing
+end
+function edge_g!(ysrc, ydst, e, vsrc, vdst, p, t)
+    ydst[1] = p[1] * sin(vsrc[1] - vdst[1])
+    ysrc[1] = -ydst[1]
+end
+EdgeModel(; f=edge_f!, g=edge_g!, dim=0, pdim=1, outdim=1)
EdgeModel :StaticEdgeM FeedForward()
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[src₊o] dst=[dst₊o]
+ └─   1 param:   [p]

This is a purely "static" edge without internal states. This means we can omit f and dim entirely. Also, we can define a variant of g without the e input

function edge_g_ff!(ysrc, ydst, vsrc, vdst, p, t)
+    ydst[1] = p[1] * sin(vsrc[1] - vdst[1])
+    ysrc[1] = -ydst[1]
+end
+EdgeModel(;g=edge_g_ff!, pdim=1, outdim=1)
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[src₊o] dst=[dst₊o]
+ └─   1 param:   [p]

which no classifies as a PureFeedForward edge. In cases like this, where the edge is actually anti symmetric we can alternatively define a single sided output function and wrapping it in an AntiSymmetric object

function edge_g_s!(ydst, vsrc, vdst, p, t)
+    ydst[1] = p[1] * sin(vsrc[1] - vdst[1])
+end
+EdgeModel(;g=AntiSymmetric(edge_g_ff!), pdim=1, outdim=1)
EdgeModel :StaticEdgeM FeedForward()
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[₋o] dst=[o]
+ └─   1 param:   [p]

which can also lead to briefer output naming. Available single sided wrappers are

Once again we can add additonal data like defining a src and dst index

function edge_g_s!(ydst, vsrc, vdst, p, t)
+    ydst[1] = p[1] * sin(vsrc[1] - vdst[1])
+end
+EdgeModel(;g=AntiSymmetric(edge_g_ff!), psym=:K=>1, outsym=:P, insym=:θ, src=1, dst=4)
EdgeModel :StaticEdgeM FeedForward() @ Edge 1=>4
+ ├─ 1/1 inputs:  src=[src₊θ] dst=[dst₊θ]
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[₋P] dst=[P]
+ └─   1 param:   [K=1]
diff --git a/previews/PR184/objects.inv b/previews/PR184/objects.inv new file mode 100644 index 00000000..86279a8c Binary files /dev/null and b/previews/PR184/objects.inv differ diff --git a/previews/PR184/search_index.js b/previews/PR184/search_index.js new file mode 100644 index 00000000..f60c051c --- /dev/null +++ b/previews/PR184/search_index.js @@ -0,0 +1,3 @@ +var documenterSearchIndex = {"docs": +[{"location":"mtk_integration/#ModelingToolkit-Integration","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"","category":"section"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"NetworkDynamics.jl is compatible with ModelingTookit.jl (MTK). The general idea is to use MTK to define component models (i.e. edge and vertex dynamics) which are then connected on network scale using NetworkDynamics.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"The main entry point for this interop are the constructors","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"VertexModel(::ODESystem, inputs, outputs)\nEdgeModel(::ODESystem, srcin, dstin, [srscout], dstout)","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"whose docstrings can be found in the Component Models with MTK section in the API.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"These constructors will:","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"transforming the states marked as input to parameters and structural_simplifying the system,\ngenerating the f and g functions,\ngenerate code for observables,\nport all supported Metadata from MTK symbols to component symbols and\noutput a Vertex-/EdgeModel function compatible with NetworkDynamics.jl.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"The main usecase for this feature is when you want to build relatively complex component models but interconnect them in a very homogeneous way (i.e. having the same output/input pairings in the whole system).","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"In theory, you can achieve everything you want to do with plain MTK. The idea of combining the two is, that NetworkDynamics offers far less flexibility when in comes to interconnection of subsystems on the network level. This might allow ND to exploit more knowledge of the structure without very expensive operations such as tearing of thousands of equations.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"warning: Warning\nModelingToolkit is a fast paced library with lots of functionality and ever growing complexity. As such the provided interface is kinda experimental. Some features of MTK are straight up unsupported, for example events within models or delay differential equations.","category":"page"},{"location":"mtk_integration/#RC-Circuit-Example","page":"ModelingToolkit Integration","title":"RC-Circuit Example","text":"","category":"section"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"In good MTK tradition, this feature will be explained along a simple RC circuit example. The Dynamic Flow in simple Gas Network example is another showcase of the MTK constructors.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"The system to model is 2 node, 1 edge network. The node output states are the voltage (to ground), the edge output sates are the currents at both ends.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"\nideal v source Resistor Capacitor\n v1 o─←────MMM────→─o v2\n │ ┴ \n (↗) ┬\n │ │\n ⏚ ⏚","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"Obviously there is no need in modeling such a small system using NetworkDynamics, however the method extends quite easily to construct large electrical networks reusing the same fundamental building blocks.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"using NetworkDynamics\nusing ModelingToolkit\nusing ModelingToolkit: t_nounits as t, D_nounits as D\nusing OrdinaryDiffEqTsit5\nusing CairoMakie","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"All our components have \"terminals\", which have a voltage and current. We don't use the @connector from MTK here because our pins mark the interface towards the network and do not follow the MTK connector semantics.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"@mtkmodel NWTerminal begin\n @variables begin\n v(t), [description=\"Voltage at node\"]\n i(t), [description=\"Current flowing into node\"]\n end\nend\nnothing #hide","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"An ideal voltage source is just a model which pins its output voltage to a fixed parameter. The source ejects whatever current is necessary. We introduce another variable i(t) to \"capture\" this current. This variable will be removed during structural simplify, but will be available for plotting through the Observables mechanism. The VertexModel can be generated from an ODESystem by providing names of the input and output states:","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"@mtkmodel VoltageSource begin\n @components begin\n p = NWTerminal()\n end\n @parameters begin\n V = 1.0\n end\n @variables begin\n i(t), [description=\"produced current by ideal voltage source (observable)\"]\n end\n @equations begin\n i ~ -p.i\n p.v ~ V\n end\nend\n@named vs = VoltageSource()\nvs_vertex = VertexModel(vs, [:p₊i], [:p₊v]; vidx=1)","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"A capacitor is a slightly more complicated model. Its voltage is defined as an differential equation based on the inflowing current.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"@mtkmodel Capacitor begin\n @components begin\n p = NWTerminal(;v=0)\n end\n @parameters begin\n C = 1.0\n end\n @equations begin\n D(p.v) ~ p.i / C\n end\nend\n@named cap = Capacitor()\ncap_vertex = VertexModel(cap, [:p₊i], [:p₊v], vidx=2)","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"For the resistor we need two pins, one for the src and one for the dst side. The equations are straight forward.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"@mtkmodel Resistor begin\n @components begin\n src = NWTerminal()\n dst = NWTerminal()\n end\n @parameters begin\n R = 1.0\n end\n @equations begin\n dst.i ~ (src.v - dst.v)/R\n src.i ~ -dst.i\n end\nend\n@named resistor = Resistor()\nresistor_edge = EdgeModel(resistor, [:src₊v], [:dst₊v], [:src₊i], [:dst₊i]; src=:vs, dst=:cap)","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"Having all those components defined, we can build the network. We don't need to provide a graph object here because we specified the placement in the graph on a per component basis.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"nw = Network([vs_vertex, cap_vertex], [resistor_edge])","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"We can see, that NetworkDynamics internally is able to reduce all of the \"output\" states. We end up with a plain ODE of a single state.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"Now we can simulate the system. For that we generate the u0 object. Since the metadata (such as default values) was automatically transferred, we can straight away construct the ODEProblem and solve the system.","category":"page"},{"location":"mtk_integration/","page":"ModelingToolkit Integration","title":"ModelingToolkit Integration","text":"u0 = NWState(nw) # generate state based on default values\nprob = ODEProblem(nw, uflat(u0), (0, 10.0), pflat(u0))\nsol = solve(prob, Tsit5())\n\n# plot the solution\nfig, ax1, p = plot(sol; idxs=VIndex(:cap, :p₊v), label=\"Capacitor Voltage\");\naxislegend(ax1)\nax2 = Axis(fig[2,1])\nplot!(ax2, sol; idxs=VIndex(:vs, :i), label=\"Current produced by ideal v source\")\naxislegend(ax2)\nfig # hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"EditURL = \"../../examples/getting_started_with_network_dynamics.jl\"","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Getting-Started","page":"Getting Started","title":"Network Diffusion","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"This introductory example explains the use of the basic types and constructors in NetworkDynamics.jl by modeling a simple diffusion on an undirected network.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"This example can be dowloaded as a normal Julia script here.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Theoretical-background","page":"Getting Started","title":"Theoretical background","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"Diffusion processes are relevant for phenomena as diverse as heat conduction, electrical currents, and random walks. Generally speaking they describe the tendency of systems to evolve towards a state of equally distributed heat, charge or concentration. In such system the local temperature (or concentration) changes according to its difference with its neighborhood, i.e. the temperature gradient.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"Let g be a graph with N nodes and adjacency matrix A. Let v = (v_1 dots v_n) be a vector of (abstract) temperatures or concentrations at each node i = 1 dots N. Then the rate of change of state v_i is described by its difference with its neighbors and we obtain the following ordinary differential equation","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"dot v_i = sum_j=1^N A_ji (v_j - v_i)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The sum on the right hand side plays the role of a (discrete) gradient. If the temperature at node i is higher than at its neighboring node j it will decrease along that edge.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Modeling-diffusion-in-NetworkDynamics.jl","page":"Getting Started","title":"Modeling diffusion in NetworkDynamics.jl","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"We begin by loading the necessary packages.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"using Graphs\nusing NetworkDynamics\nusing OrdinaryDiffEqTsit5\nusing StableRNGs\nusing Plots\nnothing #hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"From the above considerations we see that in this model the nodes do not have any internal dynamics - if a node was disconnected from the rest of the network its state would never change, since then A_ji = 0 forall j and hence dot v_i = 0. This means that the evolution of a node depends only on the interaction with its neighbors. In NetworkDynamics.jl, interactions with neighbors are described by equations for the edges.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"In order to bring this equation into the form required by NetworkDynamics.jl we need split the dynamics into edge and vertex parts and bring them into the correct input-output formulation. The vertices have one internal state v which is also the output. The input is the sum over all flows of connected edges. This directly correspons to the component model definition outlined in Mathematical Model:","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"beginaligned\ndot x^mathrmv = f^mathrm v(u^mathrm v sum_k^textincident y^mathrm e_k p^mathrm v t) = sum_k^mathrmincident y^mathrme_k \ny^mathrmv = g^mathrm v(u^mathrm v sum_k^textincident y^mathrm e_k p^mathrm v t) = x^mathrmv\nendaligned","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The edge dynamics on the other hand do not have any internal states. Thus we only define the output as the difference between the source and destination vertex:","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"beginaligned\ny^mathrm e_mathrmdst = g_mathrmdst^mathrm e(u^mathrm e y^mathrm v_mathrmsrc y^mathrm v_mathrmdst p^mathrm e t) = y^mathrm v_mathrmsrc - y^mathrm v_mathrmdst\ny^mathrm e_mathrmsrc = g_mathrmsrc^mathrm e(u^mathrm e y^mathrm v_mathrmsrc y^mathrm v_mathrmdst p^mathrm e t) = y^mathrm v_mathrmdst - y^mathrm v_mathrmsrc\nendaligned","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Definition-of-EdgeModel","page":"Getting Started","title":"Definition of EdgeModel","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"function diffusionedge_g!(e_dst, v_src, v_dst, p, t)\n # e_dst, v_src, v_dst are arrays, hence we use the broadcasting operator\n e_dst .= v_src .- v_dst\n nothing\nend\nnothing #hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The function diffusionedge_g! takes as inputs the current state of the edge e, its source vertex v_src, its destination vertex v_dst, a vector of parameters p and the time t. In order to comply with the syntax of NetworkDynamics.jl we always have to define functions for edges with exactly these arguments, even though we do not need p and t for the diffusion example.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"diffusionedge_g! is called a mutating function, since it modifies (or mutates) one of its inputs, namely the edge state e. As a convention in Julia names of mutating functions end with an !. The use of mutating functions reduces allocations and thereby speeds up computations. After the function call the edge's output value e equals the difference between its source and its destination vertex (i.e. the discrete gradient along that edge).","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"Notably, this function only models g_mathrmdst. However we can wrap this single-sided output function in an AntiSymmetric output wrapper to construct the EdgeModel:","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"nd_diffusion_edge = EdgeModel(; g=AntiSymmetric(diffusionedge_g!), outsym=[:flow])","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Definition-of-VertexModel","page":"Getting Started","title":"Definition of VertexModel","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"For undirected graphs, the edgefunction! specifies the coupling from a source- to a destination vertex. The contributions of the connected edges to a single vertex are \"aggregated\". Default aggregation is the summation of all incident edge states. The aggregated edge state is made available via the esum argument of the vertex function.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"function diffusionvertex_f!(dv, v, esum, p, t)\n # dv, v and esum are arrays, hence we use the broadcasting operator .\n dv .= esum\n nothing\nend\nnothing #hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"Just like above the input arguments v, esum, p, t are mandatory for the syntax of vertex functions. The additional input dv corresponding to the derivative of the vertex' state is mandatory for vertices described by ordinary differential equations.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The output function g is just taking part of the internal states. For that we can use the StateMask helper function g = StateMaks(1:1)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"nd_diffusion_vertex = VertexModel(; f=diffusionvertex_f!, g=StateMask(1:1), dim=1)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Constructing-the-network","page":"Getting Started","title":"Constructing the network","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"With the components defined, we can define the topology and assemble the network dynamics.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"N = 20 # number of nodes\nk = 4 # average degree\ng = barabasi_albert(N, k) # a little more exciting than a bare random graph\n\nnothing #hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The Barabási–Albert model generates a scale-free random graph.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"nd = Network(g, nd_diffusion_vertex, nd_diffusion_edge)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The constructor Network combines the component model with the topological information contained in the graph g and returns an Network compatible with the solvers of DifferentialEquations.jl.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"rng = StableRNG(1)\nx0 = randn(rng, N) # random initial conditions\node_prob = ODEProblem(nd, x0, (0.0, 2.0))\nsol = solve(ode_prob, Tsit5());\nnothing #hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"We are solving the diffusion problem on the time interval 0 2 with the Tsit5() algorithm, which is recommended by the authors of DifferentialEquations.jl for most non-stiff problems.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"plot(sol; idxs=vidxs(nd, :, :), fmt=:png)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The plotting is straightforward. The idxs keyword allows us to pass a list of indices. Indices can be also \"symbolic\" indices which specify components and their symbols directly. For example idxs = VIndex(1, :v) acesses state :v of vertex 1. See Symbolic Indexing for more details.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"In oder to collect multiple indices we can use the helper function vidxs and eidxs, which help to collect all symbolic indices matching a certain criteria.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Two-Dimensional-Extension","page":"Getting Started","title":"Two Dimensional Extension","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"To illustrate a very simple multi-dimensional case, in the following we simulate two independent diffusions on an identical graph. The first uses the symbol x and is started with initial conditions drawn from the standard normal distribution N(01), the second uses the symbol ϕ with squared standard normal inital conditions.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The symbols have to be passed with the keyword sym to VertexModel.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"N = 10 # number of nodes\nk = 4 # average degree\ng = barabasi_albert(N, k) # a little more exciting than a bare random graph\n\n# We will have two independent diffusions on the network, hence dim = 2\nnd_diffusion_vertex_2 = VertexModel(; f=diffusionvertex_f!, g=1:2, dim=2, sym=[:x, :ϕ])\nnd_diffusion_edge_2 = EdgeModel(; g=AntiSymmetric(diffusionedge_g!), outsym=[:flow_x, :flow_ϕ])\nnd_2 = Network(g, nd_diffusion_vertex_2, nd_diffusion_edge_2)\n\nx0_2 = vec(transpose([randn(rng, N) .^ 2 randn(rng, N)])) # x ~ N(0,1)^2; ϕ ~ N(0,1)\node_prob_2 = ODEProblem(nd_2, x0_2, (0.0, 3.0))\nsol_2 = solve(ode_prob_2, Tsit5());\nnothing #hide","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"Try plotting the variables ϕ_i yourself. [To write ϕ type \\phi and press TAB]","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"plot(sol_2; idxs=vidxs(nd_2, :, :x), fmt=:png)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"Using the eidxs helper function we can also plot the flow variables","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"plot(sol_2; idxs=eidxs(nd_2, :, :flow_x), fmt=:png)","category":"page"},{"location":"generated/getting_started_with_network_dynamics/#Appendix:-The-network-Laplacian-L","page":"Getting Started","title":"Appendix: The network Laplacian L","text":"","category":"section"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"The diffusion equation on a network can be rewritten as","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"dot v_i = sum_j=1^N A_ji v_j - d_i v_i = e_i^T A v - d_i v_i","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"where d_i is the degree of node i and e_i^T is the i-th standard basis vector. Introducing the diagonal matrix D that has the degree of node i in its i-th row and the Laplacian matrix L = D - A we arrive at","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"dot v = e_i^T(A - D) v","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"and finally","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"dot v = - L v","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"This is a linear system of ODEs and its solution is a matrix exponential. To study the asymptotic behaviour of the system it suffices to analyze the eigenspectrum of L. For this reason L is an important construction in network science.","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"","category":"page"},{"location":"generated/getting_started_with_network_dynamics/","page":"Getting Started","title":"Getting Started","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"EditURL = \"../../examples/stress_on_truss.jl\"","category":"page"},{"location":"generated/stress_on_truss/#Stress-on-Truss","page":"Stress on Truss","title":"Stress on Truss","text":"","category":"section"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"In this exampe we'll simulate the time evolution of a truss structure consisting of joints and stiff springs. This example can be dowloaded as a normal Julia script here.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"(Image: truss animation)","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"The mathematical model is quite simple: the vertices are point masses with positions x and y and velocities v_x and v_y. For simplicity, we add some damping to the nodal motion based on the velocity.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"beginaligned\ndotx = v_x\ndoty = v_y\ndotv_x = frac1Mleft(sum F_x - γv_xright)\ndotv_y = frac1Mleft(sum F_y - γv_yright) - g\nendaligned","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"The vertices cannot absorb any torque, so the beams only exert forces in the direction of the beam.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"F = Kcdot(L - Δd)","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"where L is the nominal lenth and Δd is the actual length of the beam.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"We start by loading the necessary packages.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"using NetworkDynamics\nusing OrdinaryDiffEqTsit5\nusing Graphs\nusing GraphMakie\nusing LinearAlgebra: norm\nusing Printf\nusing CairoMakie\nCairoMakie.activate!()","category":"page"},{"location":"generated/stress_on_truss/#Definition-of-the-dynamical-system","page":"Stress on Truss","title":"Definition of the dynamical system","text":"","category":"section"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"We need 3 models:","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"a fixed vertex which cannot change its position,\na free vertex which can move, and\na beam which connects two vertices.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"function fixed_g(pos, x, p, t)\n pos .= p\nend\nvertex_fix = VertexModel(g=fixed_g, psym=[:xfix, :yfix], outsym=[:x, :y], ff=NoFeedForward())","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"Here we need to specify the ff keyword manually, because NetworkDynamics cannot distinguish between g(out, x, p, t) (NoFeedForwarwd) and g(out, in, p, t) (PureFeedForward()) and guesses the latter when mathrmdim(x)=0.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"function free_f(dx, x, Fsum, (M, γ, g), t)\n v = view(x, 1:2)\n dx[1:2] .= (Fsum .- γ .* v) ./ M\n dx[2] -= g\n dx[3:4] .= v\n nothing\nend\nvertex_free = VertexModel(f=free_f, g=3:4, sym=[:vx=>0, :vy=>0, :x, :y],\n psym=[:M=>10, :γ=>200, :g=>9.81], insym=[:Fx, :Fy])","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"For the edge, we want to do something special. Later on, we want to color the edges according to the force they exert. Therefore, are interested in the absolut force rather than just the force vector. NetworkDynamics allows you to define so called Observed functions, which can recover additional states, so called observed, after the simulations. We can use this mechanis, to define a \"recipe\" for calculating the beam force based on the inputs (nodal positions) and the beam parameters.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"function edge_g!(F, pos_src, pos_dst, (K, L), t)\n dx = pos_dst[1] - pos_src[1]\n dy = pos_dst[2] - pos_src[2]\n d = sqrt(dx^2 + dy^2)\n Fabs = K * (L - d)\n F[1] = Fabs * dx / d\n F[2] = Fabs * dy / d\n nothing\nend\nfunction observedf(obsout, u, pos_src, pos_dst, (K, L), t)\n dx = pos_dst[1] .- pos_src[1]\n dy = pos_dst[2] .- pos_src[2]\n d = sqrt(dx^2 + dy^2)\n obsout[1] = K * (L - d)\n nothing\nend\nbeam = EdgeModel(g=AntiSymmetric(edge_g!), psym=[:K=>0.5e6, :L], outsym=[:Fx, :Fy], obsf=observedf, obssym=[:Fabs])","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"With the models define we can set up graph topology and initial positions.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"N = 5\ndx = 1.0\nshift = 0.2\ng = SimpleGraph(2*N + 1)\nfor i in 1:N\n add_edge!(g, i, i+N); add_edge!(g, i, i+N)\n if i < N\n add_edge!(g, i+1, i+N); add_edge!(g, i, i+1); add_edge!(g, i+N, i+N+1)\n end\nend\nadd_edge!(g, 2N, 2N+1)\npos0 = zeros(Point2f, 2N + 1)\npos0[1:N] = [Point((i-1)dx,0) for i in 1:N]\npos0[N+1:2*N] = [Point(i*dx + shift, 1) for i in 1:N]\npos0[2N+1] = Point(N*dx + 1, -1)\nfixed = [1,4] # set fixed vertices\nnothing #hide","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"Now can collect the vertex models and construct the Network object.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"verts = VertexModel[vertex_free for i in 1:nv(g)]\nfor i in fixed\n verts[i] = vertex_fix # use the fixed vertex for the fixed points\nend\nnw = Network(g, verts, beam)","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"In order to simulate the system we need to initialize the state and parameter vectors. Some states and parameters are shared between all vertices/edges. Those have been allready set in their constructors. The free symbols are","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"x and y for the position of the free vertices,\nxfix and yfix for the position of the fixed vertices,\nL for the nominal length of the beams.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"s = NWState(nw)\n# set x/y and xfix/yfix\nfor i in eachindex(pos0, verts)\n if i in fixed\n s.p.v[i, :xfix] = pos0[i][1]\n s.p.v[i, :yfix] = pos0[i][2]\n else\n s.v[i, :x] = pos0[i][1]\n s.v[i, :y] = pos0[i][2]\n end\nend\n# set L for edges\nfor (i,e) in enumerate(edges(g))\n s.p.e[i, :L] = norm(pos0[src(e)] - pos0[dst(e)])\nend\nnothing #hide","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"Lastly there is a special vertex at the end of the truss which has a higher mass and reduced damping.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"s.p.v[11, :M] = 200\ns.p.v[11, :γ] = 100\nnothing #hide","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"No we have everything ready to build the ODEProblem and simulate the system.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"tspan = (0.0, 12.0)\nprob = ODEProblem(nw, uflat(s), tspan, pflat(s))\nsol = solve(prob, Tsit5())\nnothing #hide","category":"page"},{"location":"generated/stress_on_truss/#Plot-the-solution","page":"Stress on Truss","title":"Plot the solution","text":"","category":"section"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"Plotting trajectories of points is kinda boring. So instead we're going to use GraphMakie.jl to create a animation of the timeseries.","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"fig = Figure(size=(1000,550));\nfig[1,1] = title = Label(fig, \"Stress on truss\", fontsize=30)\ntitle.tellwidth = false\n\nfig[2,1] = ax = Axis(fig)\nax.aspect = DataAspect();\nhidespines!(ax); # no borders\nhidedecorations!(ax); # no grid, axis, ...\nlimits!(ax, -0.1, pos0[end][1]+0.3, pos0[end][2]-0.5, 1.15) # axis limits to show full plot\n\n# get the maximum force during the simulation to get the color scale\n# It is only possible to access `:Fabs` directly becaus we've define the observable function for it!\n(fmin, fmax) = 0.3 .* extrema(Iterators.flatten(sol(sol.t, idxs=eidxs(nw, :, :Fabs))))\np = graphplot!(ax, g;\n edge_width = 4.0,\n node_size = 3*sqrt.(try s.p.v[i, :M] catch; 10.0 end for i in 1:nv(g)),\n nlabels = [i in fixed ? \"Δ\" : \"\" for i in 1:nv(g)],\n nlabels_align = (:center,:top),\n nlabels_fontsize = 30,\n elabels = [\"edge $i\" for i in 1:ne(g)],\n elabels_side = Dict(ne(g) => :right),\n edge_color = [0.0 for i in 1:ne(g)],\n edge_attr = (colorrange=(fmin,fmax),\n colormap=:diverging_bkr_55_10_c35_n256))\n\n# draw colorbar\nfig[3,1] = cb = Colorbar(fig, get_edge_plot(p), label = \"Axial force\", vertical=false)\n\nT = tspan[2]\nfps = 30\ntrange = range(0.0, sol.t[end], length=Int(T * fps))\nrecord(fig, \"truss.mp4\", trange; framerate=fps) do t\n title.text = @sprintf \"Stress on truss (t = %.2f )\" t\n s_at_t = NWState(sol, t)\n for i in eachindex(pos0)\n p[:node_pos][][i] = (s_at_t.v[i, :x], s_at_t.v[i, :y])\n end\n notify(p[:node_pos])\n load = s_at_t.e[:, :Fabs]\n p.edge_color[] = load\n p.elabels = [@sprintf(\"%.0f\", l) for l in load]\n fig\nend","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"","category":"page"},{"location":"generated/stress_on_truss/","page":"Stress on Truss","title":"Stress on Truss","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"EditURL = \"../../examples/directed_and_weighted_graphs.jl\"","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Neurodynamic-model-of-synchronization-in-the-human-brain","page":"Directed and Weighted Graphs","title":"Neurodynamic model of synchronization in the human brain","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"This example can be dowloaded as a normal Julia script here.","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Topics-covered-in-this-tutorial-include:","page":"Directed and Weighted Graphs","title":"Topics covered in this tutorial include:","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"constructing a directed, weighted graph from data\nsome useful macros\nparameter handling\nstiff equations","category":"page"},{"location":"generated/directed_and_weighted_graphs/#The-FitzHugh-Nagumo-model","page":"Directed and Weighted Graphs","title":"The FitzHugh-Nagumo model","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Dynamics of spiking neurons have been described in a simplified manner by the FitzHugh-Nagumo model.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"beginaligned\nvarepsilon dot u = u - fracu^33 - v \ndot v = u + a\nendaligned","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Here u is a fast, excitatory variable corresponding to the membrane potential and v is a slower, inhibitory varibale. varepsilon is a parameter separating these time-scales, and a is a control parameter.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"In simplified models of the brain, such relaxation oscillators may be used to model individual neurons, clusters of neurons or even larger areas in the brain. The FitzHugh-Nagumo model has been widely used for studying synchronization in neuronal activity, which in turn has been connected to physiological phenomena such as epileptic seizures.","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Coupling-relaxation-oscillators","page":"Directed and Weighted Graphs","title":"Coupling relaxation oscillators","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"While different coupling schemes for FitzHugh-Nagumo oscillators have been proposed, in this tutorial we focus on coupling of the excitatory variables via electrical gap junctions, as described by the following system of equations.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"beginaligned\nvarepsilon dot u_i = u_i - fracu_i^33 - v_i - sigma sum_j=1^N G_ij(u_i - u_j) \ndot v_i = u_i + a\nendaligned","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"This is a simple diffusive coupling mediated by the difference between activation potentials in pairs of neurons. A similar coupling term was introduced in the \"getting started\" tutorial.","category":"page"},{"location":"generated/directed_and_weighted_graphs/#The-network-topology-a-brain-atlas","page":"Directed and Weighted Graphs","title":"The network topology - a brain atlas","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"In the following we will use a directed and weighted network encoding the strength and directionality of coupling between 90 different areas of the brain [Nathalie Tzourio-Mazoyer et al., 2002, Neuroimage].","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"The network weight matrix is given as a text file containing 90 lines with 90 numbers representing the coupling strength and separated by commas ,. The data can be conveniently read into a matrix with the DelimitedFiles module.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"using DelimitedFiles\nusing SimpleWeightedGraphs, Graphs\nusing NetworkDynamics\nusing OrdinaryDiffEqTsit5\nusing OrdinaryDiffEqSDIRK\nusing StableRNGs\nusing Plots\n\n# adjust the load path for your filesystem!\nfile = joinpath(pkgdir(NetworkDynamics), \"docs\", \"examples\", \"Norm_G_DTI.txt\")\nG = readdlm(file, ',', Float64, '\\n')\nnothing #hide","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"The data structure for directed, weighted graphs is provided by the package SimpleWeightedGraphs.jl which is based on Graphs.jl.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"# First we construct a weighted, directed graph\ng_weighted = SimpleWeightedDiGraph(G)\n\n# For later use we extract the edge.weight attributes\n# . is the broadcasting operator and gets the attribute :weight for every edge\nedge_weights = getfield.(collect(edges(g_weighted)), :weight)\n\n# we promote the g_weighted graph as a directed graph (weights of the edges are included in parameters)\ng_directed = SimpleDiGraph(g_weighted)\n\nnothing #hide","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Setting-up-the-ODEProblem","page":"Directed and Weighted Graphs","title":"Setting up the ODEProblem","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Defining the VertexModel and EdgeModel is similar to the example before. The macro Base.@propagate_inbounds tells the compiler to inline the function and propagate the inbounds context. For more details see the julia documentation.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Base.@propagate_inbounds function fhn_electrical_vertex!(dv, v, esum, p, t)\n (a, ϵ) = p\n dv[1] = v[1] - v[1]^3 / 3 - v[2] + esum[1]\n dv[2] = (v[1] - a) * ϵ\n nothing\nend\nvertex = VertexModel(f=fhn_electrical_vertex!, g=1, sym=[:u, :v], psym=[:a=>0.5, :ϵ=>0.05])","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Base.@propagate_inbounds function electrical_edge!(e, v_s, v_d, (w, σ), t)\n e[1] = w * (v_s[1] - v_d[1]) * σ\n nothing\nend\nelectricaledge = EdgeModel(g=Directed(electrical_edge!), outdim=1, psym=[:weight, :σ=>0.5])","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"fhn_network! = Network(g_directed, vertex, electricaledge)","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Since this system is a directed one with thus directed edges, the keyword argument coupling is used to set the coupling of the edges to Directed().","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Parameter-handling","page":"Directed and Weighted Graphs","title":"Parameter handling","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Some of the parameters have been declared with default values. Those default values will be used when creating the NWParameter object. We can use getindex on the parameter objects to set the missing weight values.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"p = NWParameter(fhn_network!)\np.e[1:ne(g_directed), :weight] = edge_weights\nnothing #hide","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"The initial conditions could be created similarly to the parameters as an indexable NWState obejct. Since we chose a random initial condition we initialize the flat array directly:","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"x0 = randn(StableRNG(42), dim(fhn_network!)) * 5\n\nnothing #hide","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Solving-the-system","page":"Directed and Weighted Graphs","title":"Solving the system","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Now we are ready to create an ODEProblem. Since for some choices of parameters the FitzHugh-Nagumo model is stiff (i.e. numerically unstable), we use a solver with automated stiffness detection. Such a solver switches to a more stable solver only when the solution enters a region of phase space where the problem is numerically unstable. In this case we use Tsit5 and switch to TRBDF2 when necessary. AutoTsit5 is the switching version of the Tsit5 algorithm.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"Not that we call pflat on the NWParameter object to get the flat array of parameters.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"tspan = (0.0, 200.0)\nprob = ODEProblem(fhn_network!, x0, tspan, pflat(p))\nsol = solve(prob, AutoTsit5(TRBDF2()));\nnothing #hide","category":"page"},{"location":"generated/directed_and_weighted_graphs/#Plotting","page":"Directed and Weighted Graphs","title":"Plotting","text":"","category":"section"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"The plot of the excitatory variables shows that they synchronize for this choice of parameters.","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"plot(sol; idxs=vidxs(fhn_network!, :, :u), legend=false, ylim=(-5, 5), fmt=:png)","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"","category":"page"},{"location":"generated/directed_and_weighted_graphs/","page":"Directed and Weighted Graphs","title":"Directed and Weighted Graphs","text":"This page was generated using Literate.jl.","category":"page"},{"location":"initialization/#Initialization","page":"Initialization","title":"Initialization","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Initialization of the system describes the process of finding valid initial conditions, mostly a fixpoint of the system. We distinguish between two types of initialization: full system initialziation and component initialization.","category":"page"},{"location":"initialization/#Full-System-Initialization","page":"Initialization","title":"Full-System Initialization","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Full system initialization describs the process of finding a fixpoint/steady state of th entire system.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"To do so, you can use find_fixpoint, which creates a SteadyStateProblem of the whole network and tries do solve it. ","category":"page"},{"location":"initialization/#Component-wise-Initialization","page":"Initialization","title":"Component-wise Initialization","text":"","category":"section"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"In contrast to full-system initialization the goal of component-wise initialization is to find a valid initial condition for a single component first, given a network coupling.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"This can be usefull in cases, where there are nontrivial internal dynamics and states within a single vertex or edge. The idea of component-wise initialisation is to find internal states which match a given \"network coupling\" (fixed inputs and outputs).","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Lets consider the following example of a Swing-equation generator model.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"using NetworkDynamics, ModelingToolkit\nusing ModelingToolkit: t_nounits as t, D_nounits as Dt\n\n@mtkmodel Swing begin\n @variables begin\n u_r(t)=1, [description=\"bus d-voltage\", output=true]\n u_i(t)=0.1, [description=\"bus q-voltage\", output=true]\n i_r(t)=1, [description=\"bus d-current (flowing into bus)\", input=true]\n i_i(t)=0.1, [description=\"bus d-current (flowing into bus)\", input=true]\n ω(t), [guess=0.0, description=\"Rotor frequency\"]\n θ(t), [guess=0.0, description=\"Rotor angle\"]\n Pel(t), [guess=1, description=\"Electrical Power injected into the grid\"]\n end\n @parameters begin\n M=0.005, [description=\"Inertia\"]\n D=0.1, [description=\"Damping\"]\n V=sqrt(u_r^2 + u_i^2), [description=\"Voltage magnitude\"]\n ω_ref=0, [description=\"Reference frequency\"]\n Pm, [guess=0.1,description=\"Mechanical Power\"]\n end\n @equations begin\n Dt(θ) ~ ω - ω_ref\n Dt(ω) ~ 1/M * (Pm - D*ω - Pel)\n Pel ~ u_r*i_r + u_i*i_i\n u_r ~ V*cos(θ)\n u_i ~ V*sin(θ)\n end\nend\nsys = Swing(name=:swing)\nvf = VertexModel(sys, [:i_r, :i_i], [:u_r, :u_i])","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"You can see in the provided metadata, that we've set default values for the node outputs u_r, u_i, the node inputs i_r, i_i and most parameters. For some states and parameters, we've onlye provided a guess rather than a default. Variables which only have guesses are considered \"tunable\" for the initialization algorithm.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"In order to initialize the remaining variables we use initialize_component!, which is a mutating function which tries to solve the nonlinear initialization problem and store the found values for the \"free\" variables as init metadata.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"initialize_component!(vf; verbose=true)\nnothing #hide","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"vf #hide","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"Which lead to a successfull initialization of states :θ and :ω as well as parameter :Pm. To retrieve the residual you can use init_residual.","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"As a quick test we can ensure that the angle indeed matches the voltag angel:","category":"page"},{"location":"initialization/","page":"Initialization","title":"Initialization","text":"get_init(vf, :θ) ≈ atan(get_default(vf, :u_i), get_default(vf, :u_r))","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"EditURL = \"../../examples/heterogeneous_system.jl\"","category":"page"},{"location":"generated/heterogeneous_system/#Modeling-a-heterogeneous-system","page":"Heterogeneous Systems","title":"Modeling a heterogeneous system","text":"","category":"section"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"This example can be dowloaded as a normal Julia script here.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"One of the main purposes of NetworkDynamics.jl is to facilitate modeling coupled systems with heterogenities. This means that components can differ in their parameters as well as in their dynamics.","category":"page"},{"location":"generated/heterogeneous_system/#Heterogenous-parameters","page":"Heterogeneous Systems","title":"Heterogenous parameters","text":"","category":"section"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"We start by setting up a simple system of Kuramoto oscillators.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"using NetworkDynamics, OrdinaryDiffEqTsit5, Plots, Graphs\n\nN = 8\ng = watts_strogatz(N, 2, 0) # ring network\n\nfunction kuramoto_edge!(e, θ_s, θ_d, (K,), t)\n e[1] = K * sin(θ_s[1] - θ_d[1])\n nothing\nend\nedge! = EdgeModel(g=AntiSymmetric(kuramoto_edge!), outdim=1, psym=[:K=>3])","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"function kuramoto_vertex!(dθ, θ, esum, (ω0,), t)\n dθ[1] = ω0 + esum[1]\n nothing\nend\nvertex! = VertexModel(f=kuramoto_vertex!, g=StateMask(1:1), sym=[:θ], psym=[:ω0], name=:kuramoto)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"nw = Network(g, vertex!, edge!)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"To assign parameters, we can create a NWParameter object based on the nw definition. This parameter object will be pre-filled with the default parameters.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"p = NWParameter(nw)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"To set the vertex parameters, we can use indexing of the p.v field:","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"ω = collect(1:N) ./ N\nω .-= sum(ω) / N\np.v[:, :ω0] = ω\nnothing #hide","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"Here, the index pairing :, :ω is used to index state ω for all node indices.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"The parameter object contains information about the network structure. For the actual problem definition we need to throw away this wrapper and use the flat-vector representation of the parameters pflat(p). Note that pflat(p)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"Similarily, we could use NWState(nw) to create an indexable wrapper of the initial state. However in this case we can also fill create the flat state array manually:","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"x0 = collect(1:N) ./ N\nx0 .-= sum(x0) ./ N\ntspan = (0.0, 10.0)\nprob = ODEProblem(nw, x0, tspan, pflat(p))\nsol = solve(prob, Tsit5())\nplot(sol; ylabel=\"θ\", fmt=:png)","category":"page"},{"location":"generated/heterogeneous_system/#Heterogeneous-dynamics","page":"Heterogeneous Systems","title":"Heterogeneous dynamics","text":"","category":"section"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"Two paradigmatic modifications of the node model above are static nodes and nodes with inertia. A static node has no internal states and instead fixes the variable at a constant value.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"function static_g(out, u, p, t)\n out[1] = p[1]\n nothing\nend\nstatic! = VertexModel(g=static_g, outsym=[:θ], psym=[:θfix => ω[1]], name=:static)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"But wait! NetworkDynamics classified this as PureFeedForward, because it cannot distinguish between the function signatures","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"g(out, u, p, t) # PureFeedForward\ng(out, ins, p, t) # NoFeedForward","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"and since dim(u)=0 it wrongfully assumes that the latter is meant. We can overwrite the classification by passing the ff keyword:","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"static! = VertexModel(g=static_g, outsym=[:θ], psym=[:θfix => ω[1]], ff=NoFeedForward(), name=:static)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"A Kuramoto model with inertia consists of two internal variables leading to more complicated (and for many applications more realistic) local dynamics.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"function kuramoto_inertia!(dv, v, esum, (ω0,), t)\n dv[1] = v[2]\n dv[2] = ω0 - 1.0 * v[2] + esum[1]\n nothing\nend\n\ninertia! = VertexModel(f=kuramoto_inertia!, g=1:1, sym=[:θ, :ω], psym=[:ω0], name=:inertia)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"Since now we model a system with heterogeneous node dynamics we can no longer straightforwardly pass a single VertexModel to the Network constructor but instead have to hand over an Array.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"vertex_array = VertexModel[vertex! for i in 1:N]\nvertex_array[1] = static!\nvertex_array[5] = inertia! # index should correspond to the node's index in the graph\nnw_hetero! = Network(g, vertex_array, edge!)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"Now we have to take a bit more care with defining initial conditions and parameters.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"First, we can generate a NWState object based on the nw_hetero! object which will be populated with the default values.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"state = NWState(nw_hetero!)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"The node with inertia is two-dimensional, hence we need to specify two initial conditions. For the first dimension we keep the initial conditions from above and insert! another one into x0 at the correct index.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"For the θ states we will use the same initial conditins as before:","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"state.v[2:8,:θ] = x0[2:8]\nnothing #hide","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"We're still missing one initial condition: the second variable ω of the 5th vertex.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"state.v[5,:ω] = 5\nnothing #hide","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"The NWState object also contains a parameter object accessible via state.p. The edge parameters are already filled with default values. The vertex parameters can be copied from our old parmeter object p.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"state.p.v[2:8, :ω0] = p.v[2:8, :ω0]\nnothing #hide","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"For the problem construction, we need to convert the nested stuctures to flat arrays using the uflat and pflat methods.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"prob_hetero = ODEProblem(nw_hetero!, uflat(state), tspan, pflat(state))\nsol_hetero = solve(prob_hetero, Tsit5());\nnothing #hide\nplot(sol_hetero)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"For clarity we plot only the variables referring to the oscillator's angle θ and color them according to their type.","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"colors = map(vertex_array) do vertexf\n if vertexf.name == :kuramoto\n colorant\"lightseagreen\"\n elseif vertexf.name == :static\n colorant\"orange\"\n elseif vertexf.name == :inertia\n colorant\"darkred\"\n end\nend\n\nplot(sol_hetero; ylabel=\"θ\", idxs=vidxs(1:8,:θ), lc=colors', fmt=:png)","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"","category":"page"},{"location":"generated/heterogeneous_system/","page":"Heterogeneous Systems","title":"Heterogeneous Systems","text":"This page was generated using Literate.jl.","category":"page"},{"location":"metadata/#Metadata","page":"Metadata","title":"Metadata","text":"","category":"section"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Component model such as VertexModel and EdgeModel can store metadata. We distinguish between two kinds of metadata: component metadata and symbol metadata.","category":"page"},{"location":"metadata/#Component-Metadata","page":"Metadata","title":"Component Metadata","text":"","category":"section"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Component metadata is a Dict{Symbol,Any} attached to each component to store various information. Use metadata to retrieve the full dict.","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"To access the data, you can use the methods has_metadata, get_metadata and set_metadata! (see Component Metadata API).","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Special metadata: ","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":":init_residual: after Component-wise Initialization, this field stores the residual vector of the nonlinear problem.\n:graphelement: optional field to specialize the graphelement for each component (vidx) for vertices, (;src,dst) named tuple of either vertex names or vertex indices for edges. Has special accessors has_/get_/set_graphelement.","category":"page"},{"location":"metadata/#Symbol-Metadata","page":"Metadata","title":"Symbol Metadata","text":"","category":"section"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Each component stores symbol metadata. The symbol metadata is a Dict{Symbol, Dict{Symbol, Any}} which stores a metadate dict per symbol. Symbols are everything that appears in sym, psym, obssym and insym.","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"To access the data, you can use the methods has_metadata, get_metadata and set_metadata! (see Per Symbol Metadata API).","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Special cases for symbol metadata are:","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"default: Stores default values for states/parameters. In initialization, those are considered fixed.\nguess: Stores a guess for a state/parameter which needs to solved during initialization (\"free\" variables).\nbounds: Store bounds for variables/parameters\ninit: Stores the solution of the \"free\" variables during initialization.","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Fore those, there are special functions has_*, get_* and set_*!. See Per Symbol Metadata API.","category":"page"},{"location":"metadata/","page":"Metadata","title":"Metadata","text":"Those are closely aligned to the metadata use in ModelingToolkit. They are automatically copied from the ODESystem if you use MTK models to create NetworkDynamics models.","category":"page"},{"location":"symbolic_indexing/#Symbolic-Indexing","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"","category":"section"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Using SciML's SymblicIndexingInterface.jl, ND.jl provides lots of methods to access and change variables and Parameters.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"details: Setup code to make following examples work\nusing NetworkDynamics\nusing Graphs\nusing OrdinaryDiffEqTsit5\nusing Plots","category":"page"},{"location":"symbolic_indexing/#Provide-Symbol-Names","page":"Symbolic Indexing","title":"Provide Symbol Names","text":"","category":"section"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"When construction component models, you can pass symbolic names using the sym and psym keywords.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"function _edgef!(e, v_s, v_d, (K,), t)\n e .= K * (v_s[1] .- v_d[1])\nend\nedgef = EdgeModel(;g=AntiSymmetric(_edgef!), outsym=[:flow], psym=[:K=>1])","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Here we created a static diffusion edge with suitable variable and parameter names. Similarly, we define the diffusion vertex with symbolic names.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"function _vertexf!(dv, v, esum, p, t)\n dv[1] = esum[1]\nend\nvertexf = VertexModel(f=_vertexf!, g=1, sym=[:storage])","category":"page"},{"location":"symbolic_indexing/#Fundamental-Symblic-Indices","page":"Symbolic Indexing","title":"Fundamental Symblic Indices","text":"","category":"section"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"The default types for this access are the types VIndex, EIndex, VPIndex and EPIndex. Each of those symbolic indices consists of 2 elements: a reference to the network componen and a reference to the symbol within that component. As such, VIndex(2, :x) refers to variable with symbolic name :x in vertex number 2. EPIndex(4, 2) would refer to the second parameter of the edge component for the 4th edge.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"details: Setup code to make following examples work\ng = wheel_graph(5)\nnw = Network(g, vertexf, edgef)\ns = NWState(nw)\ns.v[:,:storage] .= randn(5)\nprob = ODEProblem(nw, uflat(s), (0,2), pflat(s))\nsol = solve(prob, Tsit5()) \nnothing #hide","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Those fundamental indices can be used in a lot of scenarios. Most importantly you can use them to","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"sol(sol.t; idxs=VIndex(1, :storage)) # extract timeseries out ouf solution object\nplot(sol; idxs=[VIndex(1, :storage), VIndex(5,:storage)]) # plot storage of two nodes","category":"page"},{"location":"symbolic_indexing/#Generate-Symbolic-Indices","page":"Symbolic Indexing","title":"Generate Symbolic Indices","text":"","category":"section"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Often, you need many individual symbolic indices. For that there are the helper methods vidxs, eidxs, vpidxs and epidxs. With the help of those methods you can generate arrays of symbolic indices:","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"vidxs(nw, :, :storage) # get variable \"storage\" for all nodes","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"plot(sol; idxs=vidxs(nw, :, :storage))","category":"page"},{"location":"symbolic_indexing/#NWState-and-NWParameter-Objects","page":"Symbolic Indexing","title":"NWState and NWParameter Objects","text":"","category":"section"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Internally, both state and parameters of a Network are represented using flat arrays. To access the state or parameters of a network, you can use the NWState and NWParameter objects.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"p = NWParameter(nw)","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"creates a NWParameter object for the network nw. It essentially creates a new flat parameter array and fills it with the default parameter values define in the component. The parameters in the NWParameter object can be accessed using the symbolic indices.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"p[EPIndex(5, :K)] = 2.0 # change the parameter K of the 5th edge\nnothing #hide","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Similarly, you can create a NWState object for the network nw using","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"s = NWState(nw)","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"No default values were provided in the network components, so the state array is filled with NaNs.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"s[VIndex(:, :storage)] .= randn(5) # set the (initial) storage for alle nodes \ns #hide","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"For both NWState and NWParameter objects, the there is a more convenient way to access the variables and parameters.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"@assert s.v[1, :storage] == s[VIndex(1, :storage)] # s.v -> access vertex states\n@assert s.e[1, :flow] == s[EIndex(1, :flow)] # s.e -> access edge states\n@assert s.p.e[1,:K] == p[EPIndex(1, :K)] # s.p -> access parameters","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"The NWState and NWParameter objects are mutable, thus changing them will also change the underlying wrapped flat arrays. You can allways access the flat representations by calling uflat and pflat.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"note: Note\nThe NWState and NWParameter wrappers can be constructed from various objects. Fore example, within a callback you might construct p = NWParameter(integrator) to then change the parameters of the network within the callback.","category":"page"},{"location":"symbolic_indexing/#Observables","page":"Symbolic Indexing","title":"Observables","text":"","category":"section"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Sometimes, the \"states\" you're interested in aren't really states in the DAE sense but rather algebraic derivations from DAE states, parameters and time – in accordance with the naming in the SciML-ecosystem those states are called Observables.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"A prime example of Observables are edge/vertex-outputs, such as the flow in the edge model defined above. It is also possible to define additional Observables manually by using the obssym and obsf keyword on the EdgeModel/VertexModel constructors. When building models using ModelingToolkit, the reduced algebraic states will be preserved as observables automatically.","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"Observables can be accessed like any other state, for example, the flows in the network don't show up in the state array but can be accessed in all the ways discussed above, for example","category":"page"},{"location":"symbolic_indexing/","page":"Symbolic Indexing","title":"Symbolic Indexing","text":"plot(sol; idxs=eidxs(nw, :, :flow))","category":"page"},{"location":"mathematical_model/#Mathematical-Model","page":"Mathematical Model","title":"Mathematical Model","text":"","category":"section"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The basic mathematical model of NetworkDynamics.jl splits up the system it two parts: vertex and edge components.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The main goal of NetworkDynamics.jl is, to express the overall network dynamics as a Differential-Algebraic-Equation (DAE)","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Mfracmathrmdmathrmdtu = f^mathrmnw(u p t)","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"where M is a (possibly singular) mass matrix, u is the internal state vector of the system, p are the parameters and t is the time. To make this compatible with the solvers for OrdinaryDiffEq.jl, the created Network object is a callable object","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"nw(du, u, p, t) # mutates du","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"with stored mass matrix information to build an ODEProblem based on the Network.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Instead of defining f^mathrmnw by hand, ND.jl helps you to build it automatically based on a list of decentralized nodal and edge dynamics, so-called VertexModel and EdgeModel objects. Each component model mathrm c is modeled as general input-output-system","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"beginaligned\nM_mathrm cfracmathrmdmathrmdtx_mathrm c = f^mathrm c(x^mathrm c i_mathrm c p_mathrm c t)\ny^mathrm c = g^mathrm c(x^mathrmc i_mathrm c p_mathrm c t)\nendaligned","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"where M_mathrmc is the component mass matrix, x^mathrm c are the component states, i^mathrm c are the inputs of the component and y^mathrm c is the output of the component. It is possible to have mathrmdim(x^mathrmc) = 0 and thus no internal states.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"In the network context, the output of the edges are flow variables. The outputs of vertices are potential variables. In interconnection, the flow on the edges depends on the potentials at both ends as inputs. The potentials of the nodes depend on the incoming flows from all connected edges as an input. (Here, flow and potentials are meant in a conceptional and not necessarily physical way.)","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"","category":"page"},{"location":"mathematical_model/#Vertex-Models","page":"Mathematical Model","title":"Vertex Models","text":"","category":"section"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Specifically, a (single-layer) vertex model has one input, and one output. The input is an aggregation/reduction over all incident edge outputs,","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"i^mathrm v = mathopmathrmagglimits_k^textincident y^mathrm e_k qquadtextoftenqquad\ni^mathrm v = sum_k^textincident y^mathrm e_k","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The full vertex model","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"beginaligned\nM^mathrm vfracmathrmdmathrmdtx^mathrm v = f^mathrm v(u^mathrm v i^mathrm v p^mathrm v t)\ny^mathrm v = g^mathrm v(u^mathrm v i^mathrm v p^mathrm v t)\nendaligned","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"corresponds to the Julia functions","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"function fᵥ(dxᵥ, xᵥ, e_aggr, pᵥ, t)\n # mutate dxᵥ\n nothing\nend\nfunction gᵥ(yᵥ, xᵥ, e_aggr, pᵥ, t)\n # mutate yᵥ\n nothing\nend\nvertf = VertexModel(; f=fᵥ, g=gᵥ, mass_matrix=Mᵥ, ...)","category":"page"},{"location":"mathematical_model/#Edge-Models","page":"Mathematical Model","title":"Edge Models","text":"","category":"section"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"In contrast to vertex models, edge models in general have two inputs and two outputs, both for source and destination end of the edge. We commonly use src and dst to describe the source and destination end of an edge respectively. ","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"note: On the directionality of edges\nMathematically, in a system defined on an undirected graph there is no difference between the edge (12) and (21), the edge has no direction. However, from an implementation point of view we always need to have some kind of ordering for function arguments, state order and so on. For undirected graphs, Graphs.jl chooses the direction of an edge v1->v2 such that v1 < v2.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The inputs of the edge are just the outputs of the two nodes at both ends. The output is split into two: the dst output goes to the input of the vertex at the destination end, the src output goes to the input of the vertex at the src end.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The full model of an edge","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"beginaligned\nM^mathrm efracmathrmdmathrmdtx^mathrm e = f^mathrm e(u^mathrm e y^mathrm v_mathrmsrc y^mathrm v_mathrmdst p^mathrm e t)\ny^mathrm e_mathrmdst = g_mathrmdst^mathrm e(u^mathrm e y^mathrm v_mathrmsrc y^mathrm v_mathrmdst p^mathrm e t)\ny^mathrm e_mathrmsrc = g_mathrmsrc^mathrm e(u^mathrm e y^mathrm v_mathrmsrc y^mathrm v_mathrmdst p^mathrm e t)\nendaligned","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"corresponds to the Julia functions","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"function fₑ(dxₑ, xₑ, v_src, v_dst, pₑ, t)\n # mutate dxᵥ\n nothing\nend\nfunction gₑ(y_src, y_dst, xᵥ, v_src, v_dst, pₑ, t)\n # mutate y_src and y_dst\n nothing\nend\nvertf = EdgeModel(; f=fₑ, g=gₑ, mass_matrix=Mₑ, ...)","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The sign convention for both outputs of an edge must be identical, typically, a positive flow represents a flow into the connected vertex. This is important, because the vertex only receives the flows, it does not know whether the flow was produce by the source or destination end of an edge.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":" y_src y_dst \n V_src o───←─────────→───o V_dst\n","category":"page"},{"location":"mathematical_model/#Single-Sided-Edge-Outputs","page":"Mathematical Model","title":"Single Sided Edge Outputs","text":"","category":"section"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Often, edge outputs will possess some symmetry which makes it more convenient to define \"single sided\" edge output functions","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"function g_single(y, xᵥ, v_src, v_dst, pₑ, t)\n # mutate y\n nothing\nend","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"There are multiple wrappers available to automaticially convert them into double-sided edge output functions:","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Directed(g_single) builds a double-sided function which only couples to the destination side.\nSymmetric(g_single) builds a double-sided function in which both ends receive y.\nAntiSymmetric(g_single) builds a double-sided function where the destination receives y and the source receives -y.\nFiducial(g_single_src, g_singl_dst) builds a double-sided edge output function based on two single sided functions.","category":"page"},{"location":"mathematical_model/#Feed-Forward-Behavior","page":"Mathematical Model","title":"Feed Forward Behavior","text":"","category":"section"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"The most general version of the component models can contain direct feed forwards from the input, i.e. the edge output might depend directly on the connected vertices or the vertex output might depend directly on the aggregated edge input.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Whenever possible, you should define output functions without feed forwards, i.e.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"gᵥ_noff(yᵥ, xᵥ, pᵥ, t)\ngₑ_noff([y_src,] y_dst, xᵥ, pₑ, t)","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"instead of the more general","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"gᵥ(yᵥ, xᵥ, e_aggr, pᵥ, t)\ngₑ([y_src], y_dst, xᵥ, v_src, v_dst, pₑ, t)","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"NetworkDynamics cannot couple two components with feed forward to each other. It is always possible to transform feed forward behavior to an internal state x with mass matrix entry zero to circumvent this problem. This transformation can be performed automatically by using ff_to_constraint.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"warning: Feed Forward Vertices\nAs of 11/2024, vertices with feed forward are not supported at all. Use ff_to_constraint to transform them into vertex model without FF.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"Concretely, NetworkDynamics distinguishes between 4 types of feed forward behaviors of g functions based on the FeedForwardType trait. The different types the signature of provided function g. Based on the signatures avaialable, ND.jl will try to find the correct type automaticially. Using the ff keyword in the constructors, the user can enforce a specific type.","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"PureFeedForward()","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"g!(outs..., ins..., p, t) # abstractly\ng!(out_dst, v_src, v_dst, p, t) # single-sided edge\ng!(out_src, out_dst, v_src, v_dst, p, t) # double-sided edge\ng!(v_out, e_aggr, p, t) # single layer vertex","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"FeedForward()","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"g!(outs..., x, ins..., p, t) # abstractly\ng!(out_dst, x, v_src, v_dst, p, t) # single-sided edge\ng!(out_src, out_dst, x, v_src, v_dst, p, t) # double-sided edge\ng!(v_out, x, e_aggr, p, t) # single layer vertex","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"NoFeedForward()","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"g!(outs..., x, p, t) # abstractly\ng!(out_dst, x, p, t) # single-sided edge\ng!(out_src, out_dst, x, p, t) # double-sided edge\ng!(v_out, x, p, t) # single layer vertex","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"PureStateMap()","category":"page"},{"location":"mathematical_model/","page":"Mathematical Model","title":"Mathematical Model","text":"g!(outs..., x) # abstractly\ng!(out_dst, x) # single-sided edge\ng!(out_src, out_dst, x) # double-sided edge\ng!(v_out, x) # single layer vertex","category":"page"},{"location":"API/#API","page":"API","title":"API","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"The following functions are designed for public use.","category":"page"},{"location":"API/#Network-Construction-API","page":"API","title":"Network Construction API","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"Network\ndim(::Network)\npdim(::Network)","category":"page"},{"location":"API/#NetworkDynamics.Network","page":"API","title":"NetworkDynamics.Network","text":"Network([g,] vertexf, edgef; kwarg...)\n\nConstruct a Network object from a graph g and edge and component models vertexf and edgef.\n\nArguments:\n\ng::AbstractGraph: The graph on which the network is defined. Optional, can be ommittet if all component models have a defined graphelement. See vidx and src/dst keywors for VertexModel and EdgeModel constructors respectively.\nvertexm: A single VertexModel or a vector of VertexModel objects. The order of the vertex models must mirror the order of the vertices(g) iterator.\nedgem: A single EdgeModel or a vector of EdgeModel objects. The order of the edge models must mirror the order of the edges(g) iterator.\n\nOptional keyword arguments:\n\nexecution=SequentialExecution{true}(): Execution model of the network. E.g. SequentialExecution, KAExecution, PolyesterExecution or ThreadedExecution.\naggregator=execution isa SequentialExecution ? SequentialAggregator(+) : PolyesterAggregator(+): Aggregation function applied to the edge models. E.g. SequentialAggregator, PolyesterAggregator, ThreadedAggregator, SparseAggregator.\ncheck_graphelement=true: Check if the graphelement metadata is consistent with the graph.\ndealias=false Check if the components alias eachother and create copies if necessary. This is necessary if the same component model is referenced in multiple places in the Network but you want to dynamicially asign metadata, such as initialization information to specific instances.\nverbose=false: Show additional information during construction.\n\n\n\n\n\nNetwork(nw::Network; g, vertexm, edgem, kwargs...)\n\nRebuild the Network with same graph and vertex/edge models but possibly different kwargs.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.dim-Tuple{Network}","page":"API","title":"NetworkDynamics.dim","text":"dim(nw::Network)\n\nReturns the number of dynamic states in the network, corresponts to the length of the flat state vector.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.pdim-Tuple{Network}","page":"API","title":"NetworkDynamics.pdim","text":"pdim(nw::Network)\n\nReturns the number of parameters in the network, corresponts to the length of the flat parameter vector.\n\n\n\n\n\n","category":"method"},{"location":"API/#Component-Models","page":"API","title":"Component Models","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"VertexModel()\nEdgeModel()","category":"page"},{"location":"API/#NetworkDynamics.VertexModel-Tuple{}","page":"API","title":"NetworkDynamics.VertexModel","text":"VertexModel(; kwargs...)\n\nBuild a VertexModel according to the keyword arguments.\n\nMain Arguments:\n\nf=nothing: Dynamic function of the component. Can be nothing if dim is 0.\ng: Output function of the component. Usefull helpers: StateMask\nsym/dim: Symbolic names of the states. If dim is provided, sym is set automaticially.\noutsym/outdim: Symbolic names of the outputs. If outdim is provided, outsym is set automaticially. Can be infered automaticially if g isa StateMask.\npsym/pdim=0: Symbolic names of the parameters. Ifpdimis provided,psym` is set automaticially.\nmass_matrix=I: Mass matrix of component. Can be a vector v and is then interpreted as Diagonal(v).\nname=dim>0 ? :VertexM : :StaticVertexM: Name of the component.\n\nOptional Arguments:\n\ninsym/indim: Symbolic names of the inputs. If indim is provided, insym is set automaticially.\nvidx: Index of the vertex in the graph, enables graphless constructor.\nff: FeedForwardType of component. Will be typically infered from g automaticially.\nobssym/obsf: Define additional \"observable\" states.\nsymmetadata/metadata: Provide prefilled metadata dictionaries.\n\nAll Symbol arguments can be used to set default values, i.e. psym=[:K=>1, :p].\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.EdgeModel-Tuple{}","page":"API","title":"NetworkDynamics.EdgeModel","text":"EdgeModel(; kwargs...)\n\nBuild a EdgeModel according to the keyword arguments.\n\nMain Arguments:\n\nf=nothing: Dynamic function of the component. Can be nothing if dim is 0.\ng: Output function of the component. Usefull helpers: AntiSymmetric, Symmetric, Fiducial, Directed and StateMask.\nsym/dim: Symbolic names of the states. If dim is provided, sym is set automaticially.\noutsym/outdim: Symbolic names of the outputs. If outdim is provided, outsym is set automaticially. In general, outsym for edges isa named tuple (; src, dst). However, depending on the g function, it might be enough to provide a single vector or even nothing (e.g. AntiSymmetric(StateMask(1:2))). See Building EdgeModels for examples.\npsym/pdim=0: Symbolic names of the parameters. Ifpdimis provided,psym` is set automaticially.\nmass_matrix=I: Mass matrix of component. Can be a vector v and is then interpreted as Diagonal(v).\nname=dim>0 ? :EdgeM : :StaticEdgeM: Name of the component.\n\nOptional Arguments:\n\ninsym/indim: Symbolic names of the inputs. If indim is provided, insym is set automaticially. For edges, insym is a named tuple (; src, dst). If give as vector tuple is created automaticially.\nsrc/dst: Index or name of the vertices at src and dst end. Enables graphless constructor.\nff: FeedForwardType of component. Will be typically infered from g automaticially.\nobssym/obsf: Define additional \"observable\" states.\nsymmetadata/metadata: Provide prefilled metadata dictionaries.\n\nAll Symbol arguments can be used to set default values, i.e. psym=[:K=>1, :p].\n\n\n\n\n\n","category":"method"},{"location":"API/#Component-Models-with-MTK","page":"API","title":"Component Models with MTK","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"VertexModel(::ModelingToolkit.ODESystem, ::Any, ::Any)\nEdgeModel(::ModelingToolkit.ODESystem, ::Any, ::Any, ::Any, ::Any)\nEdgeModel(::ModelingToolkit.ODESystem, ::Any, ::Any, ::Any)","category":"page"},{"location":"API/#NetworkDynamics.VertexModel-Tuple{ODESystem, Any, Any}","page":"API","title":"NetworkDynamics.VertexModel","text":"VertexModel(sys::ODESystem, inputs, outputs; ff_to_constraint=true, kwargs...)\n\nCreate a VertexModel object from a given ODESystem created with ModelingToolkit. You need to provide 2 lists of symbolic names (Symbol or Vector{Symbols}):\n\ninputs: names of variables in you equation representing the aggregated edge states\noutputs: names of variables in you equation representing the node output\n\nff_to_constraint controlls, whether output transformations g which depend on inputs should be\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.EdgeModel-Tuple{ODESystem, Vararg{Any, 4}}","page":"API","title":"NetworkDynamics.EdgeModel","text":"EdgeModel(sys::ODESystem, srcin, srcout, dstin, dstout; ff_to_constraint=false, kwargs...)\n\nCreate a EdgeModel object from a given ODESystem created with ModelingToolkit. You need to provide 4 lists of symbolic names (Symbol or Vector{Symbols}):\n\nsrcin: names of variables in you equation representing the node state at the source\ndstin: names of variables in you equation representing the node state at the destination\nsrcout: names of variables in you equation representing the output at the source\ndstout: names of variables in you equation representing the output at the destination\n\nff_to_constraint controlls, whether output transformations g which depend on inputs should be transformed into constraints.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.EdgeModel-Tuple{ODESystem, Any, Any, Any}","page":"API","title":"NetworkDynamics.EdgeModel","text":"EdgeModel(sys::ODESystem, srcin, dstin, AntiSymmetric(dstout); ff_to_constraint=false, kwargs...)\n\nCreate a EdgeModel object from a given ODESystem created with ModelingToolkit.\n\nHere you only need to provide one list of output symbols: dstout. To make it clear how to handle the single-sided output definiton, you musst wrap the symbol vector in\n\nAntiSymmetric(dstout),\nSymmetric(dstout), or\nDirected(dstout).\n\nff_to_constraint controlls, whether output transformations g which depend on inputs should be transformed into constraints.\n\n\n\n\n\n","category":"method"},{"location":"API/#Output-Function-Helpers/Wrappers","page":"API","title":"Output Function Helpers/Wrappers","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"StateMask\nSymmetric\nAntiSymmetric\nDirected\nFiducial","category":"page"},{"location":"API/#NetworkDynamics.StateMask","page":"API","title":"NetworkDynamics.StateMask","text":"StateMask(i::AbstractArray)\nStateMaks(i::Number)\n\nA StateMask is a predefined output function. It can be used to define the output of a component model by picking from the internal state.\n\nI.e. g=StateMask(2:3) in a vertex function will output the internal states 2 and 3. In many contexts, StateMasks can be constructed implicitly by just providing the indices, e.g. g=1:2.\n\nFor EdgeModel this needs to be combined with a Directed, Symmetric, AntiSymmetric or Fiducial coupling, e.g. g=Fiducial(1:2, 3:4) forwards states 1:2 to dst and states 3:4 to src.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.Symmetric","page":"API","title":"NetworkDynamics.Symmetric","text":"Symmetric(g)\n\nWraps a single-sided output function g turns it into a double sided output function which applies\n\ny_dst = g(...)\ny_src = y_dst\n\ng can be a Number/AbstractArray to impicitly wrap the corresponding StateMask.\n\nSee also AntiSymmetric, Directed, Fiducial and StateMask.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.AntiSymmetric","page":"API","title":"NetworkDynamics.AntiSymmetric","text":"AntiSymmetric(g_dst)\n\nWraps a single-sided output function g_dst turns it into a double sided output function which applies\n\ny_dst = g_dst(...)\ny_src = -y_dst\n\ng_dst can be a Number/AbstractArray to impicitly wrap the corresponding StateMask.\n\nSee also Symmetric, Directed, Fiducial and StateMask.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.Directed","page":"API","title":"NetworkDynamics.Directed","text":"Directed(g_dst)\n\nWraps a single-sided output function g_dst turns it into a double sided output function which applies\n\ny_dst = g_dst(...)\n\nWith Directed there is no output for the src side. g_dst can be a Number/AbstractArray to impicitly wrap the corresponding StateMask.\n\nSee also AntiSymmetric, Symmetric, Fiducial and StateMask.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.Fiducial","page":"API","title":"NetworkDynamics.Fiducial","text":"Fiducial(g_src, g_dst)\n\nWraps two single-sided output function g_src and g_dst and turns them into a double sided output function which applies\n\ny_dst = g_src(...)\ny_src = g_dst(...)\n\ng can be a Number/AbstractArray to impicitly wrap the corresponding StateMask.\n\nSee also AntiSymmetric, Directed, Fiducial and StateMask.\n\n\n\n\n\n","category":"type"},{"location":"API/#Accessors-for-Component-Properties","page":"API","title":"Accessors for Component Properties","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"fftype\ndim(::NetworkDynamics.ComponentModel)\nsym\noutdim\noutsym\npdim(::NetworkDynamics.ComponentModel)\npsym\nobssym\nhasinsym\ninsym\nhasindim\nindim","category":"page"},{"location":"API/#NetworkDynamics.fftype","page":"API","title":"NetworkDynamics.fftype","text":"fftype(x)\n\nRetrieve the feed forward trait of x.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.dim-Tuple{NetworkDynamics.ComponentModel}","page":"API","title":"NetworkDynamics.dim","text":"dim(c::ComponentModel)::Int\n\nRetrieve the dimension of the component.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.sym","page":"API","title":"NetworkDynamics.sym","text":"sym(c::ComponentModel)::Vector{Symbol}\n\nRetrieve the symbols of the component.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.outdim","page":"API","title":"NetworkDynamics.outdim","text":"outdim(c::VertexModel)::Int\noutdim(c::EdgeModel)::@NamedTuple(src::Int, dst::Int)\n\nRetrieve the output dimension of the component\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.outsym","page":"API","title":"NetworkDynamics.outsym","text":"outsym(c::VertexModel)::Vector{Symbol} outsym(c::EdgeModel)::@NamedTuple{src::Vector{Symbol}, dst::Vector{Symbol}}\n\nRetrieve the output symbols of the component.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.pdim-Tuple{NetworkDynamics.ComponentModel}","page":"API","title":"NetworkDynamics.pdim","text":"pdim(c::ComponentModel)::Int\n\nRetrieve the parameter dimension of the component.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.psym","page":"API","title":"NetworkDynamics.psym","text":"psym(c::ComponentModel)::Vector{Symbol}\n\nRetrieve the parameter symbols of the component.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.obssym","page":"API","title":"NetworkDynamics.obssym","text":"obssym(c::ComponentModel)::Vector{Symbol}\n\nRetrieve the observation symbols of the component.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.hasinsym","page":"API","title":"NetworkDynamics.hasinsym","text":"hasinsym(c::ComponentModel)\n\nChecks if the optioan field insym is present in the component model.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.insym","page":"API","title":"NetworkDynamics.insym","text":"insym(c::VertexModel)::Vector{Symbol}\ninsym(c::EdgeModel)::@NamedTuple{src::Vector{Symbol}, dst::Vector{Symbol}}\n\nMusst be called after hasinsym/hasindim returned true. Gives the insym vector(s). For vertex model just a single vector, for edges it returns a named tuple (; src, dst) with two symbol vectors.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.hasindim","page":"API","title":"NetworkDynamics.hasindim","text":"hasindim(c::ComponentModel)\n\nChecks if the optioan field insym is present in the component model.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.indim","page":"API","title":"NetworkDynamics.indim","text":"indim(c::VertexModel)::Int\nindim(c::EdgeModel)::@NamedTuple{src::Int,dst::Int}\n\nMusst be called after hasinsym/hasindim returned true. Gives the input dimension(s).\n\n\n\n\n\n","category":"function"},{"location":"API/#FeedForwardType-Traits","page":"API","title":"FeedForwardType-Traits","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"FeedForwardType\nPureFeedForward\nFeedForward\nNoFeedForward\nPureStateMap","category":"page"},{"location":"API/#NetworkDynamics.FeedForwardType","page":"API","title":"NetworkDynamics.FeedForwardType","text":"abstract type FeedForwardType end\n\nAbstract supertype for the FeedForwardType traits.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.PureFeedForward","page":"API","title":"NetworkDynamics.PureFeedForward","text":"PureFeedForward <: FeedForwardType\n\nTrait for component output functions g that have pure feed forward behavior (do not depend on x):\n\ng!(outs..., ins..., p, t)\n\nSee also FeedForward, NoFeedForward and PureStateMap.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.FeedForward","page":"API","title":"NetworkDynamics.FeedForward","text":"FeedForward <: FeedForwardType\n\nTrait for component output functions g that have feed forward behavior. May depend on everything:\n\ng!(outs..., x, ins..., p, t)\n\nSee also PureFeedForward, NoFeedForward and PureStateMap.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.NoFeedForward","page":"API","title":"NetworkDynamics.NoFeedForward","text":"NoFeedForward <: FeedForwardType\n\nTrait for component output functions g that have no feed forward behavior (do not depend on inputs):\n\ng!(outs..., x, p, t)\n\nSee also PureFeedForward, FeedForward and PureStateMap.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.PureStateMap","page":"API","title":"NetworkDynamics.PureStateMap","text":"PureStateMap <: FeedForwardType\n\nTrait for component output functions g that only depends on state:\n\ng!(outs..., x)\n\nSee also PureFeedForward, FeedForward and NoFeedForward.\n\n\n\n\n\n","category":"type"},{"location":"API/#Symbolic-Indexing-API","page":"API","title":"Symbolic Indexing API","text":"","category":"section"},{"location":"API/#Network-Parameter-Object","page":"API","title":"Network Parameter Object","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"NWParameter\nNWParameter(::Any)\nNWParameter(::NWParameter)\nNWParameter(::SciMLBase.DEIntegrator)","category":"page"},{"location":"API/#NetworkDynamics.NWParameter","page":"API","title":"NetworkDynamics.NWParameter","text":"NWParameter(nw_or_nw_wraper, pflat)\n\nIndexable wrapper for flat parameter array pflat. Needs Network or wrapper of Network, e.g. ODEProblem.\n\np = NWParameter(nw)\np.v[idx, :sym] # get parameter :sym of vertex idx\np.e[idx, :sym] # get parameter :sym of edge idx\np[s::Union{VPIndex, EPIndex}] # get parameter for specific index\n\nGet flat array representation using pflat(p).\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.NWParameter-Tuple{Any}","page":"API","title":"NetworkDynamics.NWParameter","text":"NWParameter(nw_or_nw_wraper;\n ptype=Vector{Float64}, pfill=filltype(ptype), default=true)\n\nCreates \"empty\" NWParameter object for the Network/Wrapper nw with flat type ptype. The array will be prefilled with pfill (defaults to NaN).\n\nIf default=true the default parameter values attached to the network components will be loaded.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.NWParameter-Tuple{NWParameter}","page":"API","title":"NetworkDynamics.NWParameter","text":"NWParameter(p::NWParameter; ptype=typeof(p.pflat))\n\nCreate NWParameter based on other parameter object, just convert type.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.NWParameter-Tuple{SciMLBase.DEIntegrator}","page":"API","title":"NetworkDynamics.NWParameter","text":"NWParameter(int::SciMLBase.DEIntegrator)\n\nCreate NWParameter object from integrator.\n\n\n\n\n\n","category":"method"},{"location":"API/#Network-State-Object","page":"API","title":"Network State Object","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"NWState\nNWState(::Any)\nNWState(::NWState)\nNWState(::NWParameter)\nNWState(::SciMLBase.DEIntegrator)\nuflat\npflat","category":"page"},{"location":"API/#NetworkDynamics.NWState","page":"API","title":"NetworkDynamics.NWState","text":"NWState(nw_or_nw_wrapper, uflat, [pflat], [t])\n\nIndexable wrapper for flat state & parameter array. Needs Network or wrapper of Network, e.g. ODEProblem.\n\ns = NWState(nw)\ns.v[idx, :sym] # get state :sym of vertex idx\ns.e[idx, :sym] # get state :sym of edge idx\ns.p.v[idx, :sym] # get parameter :sym of vertex idx\ns.p.e[idx, :sym] # get parameter :sym of edge idx\ns[s::Union{VIndex, EIndex, EPIndex, VPIndex}] # get parameter for specific index\n\nGet flat array representation using uflat(s) and pflat(s).\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.NWState-Tuple{Any}","page":"API","title":"NetworkDynamics.NWState","text":"NWState(nw_or_nw_wrapper;\n utype=Vector{Float64}, ufill=filltype(utype),\n ptype=Vector{Float64}, pfill=filltype(ptype), default=true)\n\nCreates \"empty\" NWState object for the Network/Wrapper nw with flat types utype & ptype. The arrays will be prefilled with ufill and pfill respectively (defaults to NaN).\n\nIf default=true the default state & parameter values attached to the network components will be loaded.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.NWState-Tuple{NWState}","page":"API","title":"NetworkDynamics.NWState","text":"NWState(p::NWState; utype=typeof(uflat(s)), ptype=typeof(pflat(s)))\n\nCreate NWState based on other state object, just convert types.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.NWState-Tuple{NWParameter}","page":"API","title":"NetworkDynamics.NWState","text":"NWState(p::NWParameter; utype=Vector{Float64}, ufill=filltype(utype), default=true)\n\nCreate NWState based on existing NWParameter object.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.NWState-Tuple{SciMLBase.DEIntegrator}","page":"API","title":"NetworkDynamics.NWState","text":"NWState(int::SciMLBase.DEIntegrator)\n\nCreate NWState object from integrator.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.uflat","page":"API","title":"NetworkDynamics.uflat","text":"uflat(s::NWState)\n\nRetrieve the wrapped flat array representation of the state.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.pflat","page":"API","title":"NetworkDynamics.pflat","text":"pflat(p::NWParameter)\npflat(s::NWState)\n\nRetrieve the wrapped flat array representation of the parameters.\n\n\n\n\n\n","category":"function"},{"location":"API/#Symbolic-Indices","page":"API","title":"Symbolic Indices","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"VIndex\nEIndex\nVPIndex\nEPIndex","category":"page"},{"location":"API/#NetworkDynamics.VIndex","page":"API","title":"NetworkDynamics.VIndex","text":"VIndex{C,S} <: SymbolicStateIndex{C,S}\nidx = VIndex(comp, sub)\n\nA symbolic index for a vertex state variable.\n\ncomp: the component index, either int or a collection of ints\nsub: the subindex, either int, symbol or a collection of those.\n\nVIndex(1, :P) # vertex 1, variable :P\nVIndex(1:5, 1) # first state of vertices 1 to 5\nVIndex(7, (:x,:y)) # states :x and :y of vertex 7\n\nCan be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWState, NWParameter or ODESolution.\n\nSee also: EIndex, VPIndex, EPIndex\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.EIndex","page":"API","title":"NetworkDynamics.EIndex","text":"EIndex{C,S} <: SymbolicStateIndex{C,S}\nidx = EIndex(comp, sub)\n\nA symbolic index for an edge state variable.\n\ncomp: the component index, either int or a collection of ints\nsub: the subindex, either int, symbol or a collection of those.\n\nEIndex(1, :P) # edge 1, variable :P\nEIndex(1:5, 1) # first state of edges 1 to 5\nEIndex(7, (:x,:y)) # states :x and :y of edge 7\n\nCan be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWState, NWParameter or ODESolution.\n\nSee also: VIndex, VPIndex, EPIndex\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.VPIndex","page":"API","title":"NetworkDynamics.VPIndex","text":"VPIndex{C,S} <: SymbolicStateIndex{C,S}\nidx = VPIndex(comp, sub)\n\nA symbolic index into the parameter a vertex:\n\ncomp: the component index, either int or a collection of ints\nsub: the subindex, either int, symbol or a collection of those.\n\nCan be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWParameter or ODEProblem.\n\nSee also: EPIndex, VIndex, EIndex\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.EPIndex","page":"API","title":"NetworkDynamics.EPIndex","text":"EPIndex{C,S} <: SymbolicStateIndex{C,S}\nidx = VEIndex(comp, sub)\n\nA symbolic index into the parameter a vertex:\n\ncomp: the component index, either int or a collection of ints\nsub: the subindex, either int, symbol or a collection of those.\n\nCan be used to index into objects supporting the SymbolicIndexingInterface, e.g. NWParameter or ODEProblem.\n\nSee also: VPIndex, VIndex, EIndex\n\n\n\n\n\n","category":"type"},{"location":"API/#Index-generators","page":"API","title":"Index generators","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"vidxs\neidxs\nvpidxs\nepidxs","category":"page"},{"location":"API/#NetworkDynamics.vidxs","page":"API","title":"NetworkDynamics.vidxs","text":"vidxs([inpr], components=:, variables=:) :: Vector{VIndex}\n\nGenerate vector of symbolic indexes for vertices.\n\ninpr: Only needed for name matching or : access. Can be Network, sol, prob, ...\ncomponents: Number/Vector, :, Symbol (name matches), String/Regex (name contains)\nvariables: Symbol/Number/Vector, :, String/Regex (all sym containing)\n\nExamples:\n\nvidxs(nw) # all vertex state indices\nvidxs(1:2, :u) # [VIndex(1, :u), VIndex(2, :u)]\nvidxs(nw, :, [:u, :v]) # [VIndex(i, :u), VIndex(i, :v) for i in 1:nv(nw)]\nvidxs(nw, \"ODEVertex\", :) # all symbols of all vertices with name containing \"ODEVertex\"\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.eidxs","page":"API","title":"NetworkDynamics.eidxs","text":"vidxs([inpr], components=:, variables=:) :: Vector{EIndex}\n\nGenerate vector of symbolic indexes for edges.\n\ninpr: Only needed for name matching or : access. Can be Network, sol, prob, ...\ncomponents: Number/Vector, :, Symbol (name matches), String/Regex (name contains)\nvariables: Symbol/Number/Vector, :, String/Regex (all sym containing)\n\nExamples:\n\neidxs(nw) # all edge state indices\neidxs(1:2, :u) # [EIndex(1, :u), EIndex(2, :u)]\neidxs(nw, :, [:u, :v]) # [EIndex(i, :u), EIndex(i, :v) for i in 1:ne(nw)]\neidxs(nw, \"FlowEdge\", :) # all symbols of all edges with name containing \"FlowEdge\"\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.vpidxs","page":"API","title":"NetworkDynamics.vpidxs","text":"vpidxs([inpr], components=:, variables=:) :: Vector{VPIndex}\n\nGenerate vector of symbolic indexes for parameters. See vidxs for more information.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.epidxs","page":"API","title":"NetworkDynamics.epidxs","text":"epidxs([inpr], components=:, variables=:) :: Vector{EPIndex}\n\nGenerate vector of symbolic indexes for parameters. See eidxs for more information.\n\n\n\n\n\n","category":"function"},{"location":"API/#Metadata-API","page":"API","title":"Metadata API","text":"","category":"section"},{"location":"API/#Component-Metadata-API","page":"API","title":"Component Metadata API","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"metadata\nhas_metadata(::NetworkDynamics.ComponentModel, ::Symbol)\nget_metadata(::NetworkDynamics.ComponentModel, ::Symbol)\nset_metadata!(::NetworkDynamics.ComponentModel, ::Symbol, ::Any)\nhas_graphelement\nget_graphelement\nset_graphelement!","category":"page"},{"location":"API/#NetworkDynamics.metadata","page":"API","title":"NetworkDynamics.metadata","text":"metadata(c::ComponentModel)\n\nRetrieve metadata object for the component.\n\nSee also metadata\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.has_metadata-Tuple{NetworkDynamics.ComponentModel, Symbol}","page":"API","title":"NetworkDynamics.has_metadata","text":"has_metadata(c::ComponentModel, key::Symbol)\n\nChecks if metadata key is present for the component.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.get_metadata-Tuple{NetworkDynamics.ComponentModel, Symbol}","page":"API","title":"NetworkDynamics.get_metadata","text":"get_metadata(c::ComponentModel, key::Symbol)\n\nRetrieves the metadata key for the component.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.set_metadata!-Tuple{NetworkDynamics.ComponentModel, Symbol, Any}","page":"API","title":"NetworkDynamics.set_metadata!","text":"set_metadata!(c::ComponentModel, key::Symbol, value)\n\nSets the metadata key for the component to value.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.has_graphelement","page":"API","title":"NetworkDynamics.has_graphelement","text":"has_graphelement(c)\n\nChecks if the edge or vetex function function has the graphelement metadata.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.get_graphelement","page":"API","title":"NetworkDynamics.get_graphelement","text":"get_graphelement(c::EdgeModel)::@NamedTuple{src::T, dst::T}\nget_graphelement(c::VertexModel)::Int\n\nRetrieves the graphelement metadata for the component model. For edges this returns a named tupe (;src, dst) where both are either integers (vertex index) or symbols (vertex name).\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.set_graphelement!","page":"API","title":"NetworkDynamics.set_graphelement!","text":"set_graphelement!(c::EdgeModel, src, dst)\nset_graphelement!(c::VertexModel, vidx)\n\nSets the graphelement metadata for the edge model. For edges this takes two arguments src and dst which are either integer (vertex index) or symbol (vertex name). For vertices it takes a single integer vidx.\n\n\n\n\n\n","category":"function"},{"location":"API/#Per-Symbol-Metadata-API","page":"API","title":"Per-Symbol Metadata API","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"symmetadata\nget_metadata(::NetworkDynamics.ComponentModel, ::Symbol, ::Symbol)\nhas_metadata(::NetworkDynamics.ComponentModel, ::Symbol, ::Symbol)\nset_metadata!(::NetworkDynamics.ComponentModel, ::Symbol, ::Symbol, ::Any)\nhas_default\nget_default\nset_default!\nhas_guess\nget_guess\nset_guess!\nhas_init\nget_init\nset_init!\nhas_bounds\nget_bounds\nset_bounds!","category":"page"},{"location":"API/#NetworkDynamics.symmetadata","page":"API","title":"NetworkDynamics.symmetadata","text":"symmetadata(c::ComponentModel)::Dict{Symbol,Dict{Symbol,Any}}\n\nRetrieve the metadata dictionary for the symbols. Keys are the names of the symbols as they appear in sym, psym, obssym and insym.\n\nSee also symmetadata\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.get_metadata-Tuple{NetworkDynamics.ComponentModel, Symbol, Symbol}","page":"API","title":"NetworkDynamics.get_metadata","text":"get_metadata(c::ComponentModel, sym::Symbol, key::Symbol)\n\nRetrievs the metadata key for symbol sym.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.has_metadata-Tuple{NetworkDynamics.ComponentModel, Symbol, Symbol}","page":"API","title":"NetworkDynamics.has_metadata","text":"has_metadata(c::ComponentModel, sym::Symbol, key::Symbol)\n\nChecks if symbol metadata key is present for symbol sym.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.set_metadata!-Tuple{NetworkDynamics.ComponentModel, Symbol, Symbol, Any}","page":"API","title":"NetworkDynamics.set_metadata!","text":"set_metadata!(c::ComponentModel, sym::Symbol, key::Symbol, value)\nset_metadata!(c::ComponentModel, sym::Symbol, pair)\n\nSets the metadata key for symbol sym to value.\n\n\n\n\n\n","category":"method"},{"location":"API/#NetworkDynamics.has_default","page":"API","title":"NetworkDynamics.has_default","text":"has_default(c::ComponentModel, sym::Symbol)\n\nChecks if a default value is present for symbol sym.\n\nSee also get_default, set_default!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.get_default","page":"API","title":"NetworkDynamics.get_default","text":"get_default(c::ComponentModel, sym::Symbol)\n\nReturns the default value for symbol sym.\n\nSee also has_default, set_default!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.set_default!","page":"API","title":"NetworkDynamics.set_default!","text":"set_default(c::ComponentModel, sym::Symbol, value)\n\nSets the default value for symbol sym to value.\n\nSee also has_default, get_default.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.has_guess","page":"API","title":"NetworkDynamics.has_guess","text":"has_guess(c::ComponentModel, sym::Symbol)\n\nChecks if a guess value is present for symbol sym.\n\nSee also get_guess, set_guess!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.get_guess","page":"API","title":"NetworkDynamics.get_guess","text":"get_guess(c::ComponentModel, sym::Symbol)\n\nReturns the guess value for symbol sym.\n\nSee also has_guess, set_guess!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.set_guess!","page":"API","title":"NetworkDynamics.set_guess!","text":"set_guess(c::ComponentModel, sym::Symbol, value)\n\nSets the guess value for symbol sym to value.\n\nSee also has_guess, get_guess.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.has_init","page":"API","title":"NetworkDynamics.has_init","text":"has_init(c::ComponentModel, sym::Symbol)\n\nChecks if a init value is present for symbol sym.\n\nSee also get_init, set_init!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.get_init","page":"API","title":"NetworkDynamics.get_init","text":"get_init(c::ComponentModel, sym::Symbol)\n\nReturns the init value for symbol sym.\n\nSee also has_init, set_init!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.set_init!","page":"API","title":"NetworkDynamics.set_init!","text":"set_init(c::ComponentModel, sym::Symbol, value)\n\nSets the init value for symbol sym to value.\n\nSee also has_init, get_init.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.has_bounds","page":"API","title":"NetworkDynamics.has_bounds","text":"has_bounds(c::ComponentModel, sym::Symbol)\n\nChecks if a bounds value is present for symbol sym.\n\nSee also get_bounds, set_bounds!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.get_bounds","page":"API","title":"NetworkDynamics.get_bounds","text":"get_bounds(c::ComponentModel, sym::Symbol)\n\nReturns the bounds value for symbol sym.\n\nSee also has_bounds, set_bounds!.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.set_bounds!","page":"API","title":"NetworkDynamics.set_bounds!","text":"set_bounds(c::ComponentModel, sym::Symbol, value)\n\nSets the bounds value for symbol sym to value.\n\nSee also has_bounds, get_bounds.\n\n\n\n\n\n","category":"function"},{"location":"API/#Initialization","page":"API","title":"Initialization","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"find_fixpoint\ninitialize_component!\ninit_residual","category":"page"},{"location":"API/#NetworkDynamics.find_fixpoint","page":"API","title":"NetworkDynamics.find_fixpoint","text":"find_fixpoint(nw::Network, [x0::NWState=NWState(nw)], [p::NWParameter=x0.p]; kwargs...)\nfind_fixpoint(nw::Network, x0::AbstractVector, p::AbstractVector; kwargs...)\nfind_fixpoint(nw::Network, x0::AbstractVector; kwargs...)\n\nConvenience wrapper around SteadyStateProblem from SciML-ecosystem. Constructs and solves the steady state problem, returns found value wrapped as NWState.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.initialize_component!","page":"API","title":"NetworkDynamics.initialize_component!","text":"initialize_component!(cf::ComponentModel; verbose=true, kwargs...)\n\nInitialize a ComponentModel by solving the corresponding NonlinearLeastSquaresProblem. During initialization, everyting which has a default value (see Metadata) is considered \"fixed\". All other variables are considered \"free\" and are solved for. The initial guess for each variable depends on the guess value in the Metadata.\n\nThe result is stored in the ComponentModel itself. The values of the free variables are stored in the metadata field init.\n\nThe kwargs are passed to the nonlinear solver.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.init_residual","page":"API","title":"NetworkDynamics.init_residual","text":"init_residual(cf::T; t=NaN, recalc=false)\n\nCalculates the residual |du| for the given component model for the values provided via default and init Metadata.\n\nIf recalc=false just return the residual determined in the actual initialization process.\n\nSee also initialize_component!.\n\n\n\n\n\n","category":"function"},{"location":"API/#Execution-Types","page":"API","title":"Execution Types","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"ExecutionStyle\nSequentialExecution\nPolyesterExecution\nThreadedExecution\nKAExecution","category":"page"},{"location":"API/#NetworkDynamics.ExecutionStyle","page":"API","title":"NetworkDynamics.ExecutionStyle","text":"abstract type ExecutionStyle{buffered::Bool} end\n\nAbstract type for execution style. The coreloop dispatches based on the Execution style stored in the network object.\n\nbuffered=true means that the edge input es explicitly gathered, i.e. the vertex outputs in the output buffer will be copied into a dedicated input buffer for the edges.\nbuffered=false means, that the edge inputs are not explicitly gathered, but the corloop will perform a redirected lookup into the output buffer.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.SequentialExecution","page":"API","title":"NetworkDynamics.SequentialExecution","text":"struct SequentialExecution{buffered::Bool}\n\nSequential execution, no parallelism. For buffered see ExecutionStyle.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.PolyesterExecution","page":"API","title":"NetworkDynamics.PolyesterExecution","text":"struct PolyesterExecution{buffered}\n\nParallel execution using Polyester.jl. For buffered see ExecutionStyle.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.ThreadedExecution","page":"API","title":"NetworkDynamics.ThreadedExecution","text":"struct ThreadedExecution{buffered}\n\nParallel execution using Julia threads. For buffered see ExecutionStyle.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.KAExecution","page":"API","title":"NetworkDynamics.KAExecution","text":"struct KAExecution{buffered}\n\nParallel execution using KernelAbstractions.jl. Works with GPU and CPU arrays. For buffered see ExecutionStyle.\n\n\n\n\n\n","category":"type"},{"location":"API/#Aggregators","page":"API","title":"Aggregators","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"Aggregator\nSequentialAggregator\nSparseAggregator\nThreadedAggregator\nPolyesterAggregator\nKAAggregator","category":"page"},{"location":"API/#NetworkDynamics.Aggregator","page":"API","title":"NetworkDynamics.Aggregator","text":"abstract type Aggregator end\n\nAbstract sypertype for aggregators. Aggregators operate on the output buffer of all components and fill the aggregation buffer with the aggregatated edge values per vertex.\n\nAll aggregators have the constructor\n\nAggegator(aggfun)\n\nfor example\n\nSequentialAggreator(+)\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.SequentialAggregator","page":"API","title":"NetworkDynamics.SequentialAggregator","text":"SequentialAggregator(aggfun)\n\nSequential aggregation.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.SparseAggregator","page":"API","title":"NetworkDynamics.SparseAggregator","text":"SparseAggregator(+)\n\nOnly works with additive aggregation +. Aggregates via sparse inplace matrix multiplication. Works with GPU Arrays.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.ThreadedAggregator","page":"API","title":"NetworkDynamics.ThreadedAggregator","text":"ThreadedAggregator(aggfun)\n\nParallel aggregation using Julia threads.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.PolyesterAggregator","page":"API","title":"NetworkDynamics.PolyesterAggregator","text":"PolyesterAggregator(aggfun)\n\nParallel aggregation using Polyester.jl.\n\n\n\n\n\n","category":"type"},{"location":"API/#NetworkDynamics.KAAggregator","page":"API","title":"NetworkDynamics.KAAggregator","text":"KAAggregator(aggfun)\n\nParallel aggregation using KernelAbstractions.jl. Works with both GPU and CPU arrays.\n\n\n\n\n\n","category":"type"},{"location":"API/#Utils","page":"API","title":"Utils","text":"","category":"section"},{"location":"API/","page":"API","title":"API","text":"save_parameters!\nff_to_constraint\nBase.copy(::NetworkDynamics.ComponentModel)","category":"page"},{"location":"API/#NetworkDynamics.save_parameters!","page":"API","title":"NetworkDynamics.save_parameters!","text":"save_parameters!(integrator::SciMLBase.DEIntegrator)\n\nSave the current parameter values in the integrator. Call this function inside callbacks if the parameter values have changed. This will store a timeseries of said parameters in the solution object, thus alowing us to recosntruct observables which depend on time-dependet variables.\n\n\n\n\n\n","category":"function"},{"location":"API/#NetworkDynamics.ff_to_constraint","page":"API","title":"NetworkDynamics.ff_to_constraint","text":"ff_to_constraint(v::VertexModel)\n\nTakes VertexModel v with feed forward and turns all algebraic output states into internal states by defining algebraic constraints contraints 0 = out - g(...). The new output function is just a StateMask into the extended internal state vector.\n\nReturns the transformed VertexModel.\n\n\n\n\n\n","category":"function"},{"location":"API/#Base.copy-Tuple{NetworkDynamics.ComponentModel}","page":"API","title":"Base.copy","text":"copy(c::NetworkDynamics.ComponentModel)\n\nShallow copy of the component model. Creates a deepcopy of metadata and symmetadata but references the same objects everywhere else.\n\n\n\n\n\n","category":"method"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"EditURL = \"../../examples/cascading_failure.jl\"","category":"page"},{"location":"generated/cascading_failure/#Cascading-Failure","page":"Cascading Failure","title":"Cascading Failure","text":"","category":"section"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"This script reimplements the minimal example of a dynamic cascading failure described in Schäfer et al. (2018) [1]. In is an example how to use callback functions to change network parameters. In this case to disable certain lines. This script can be dowloaded as a normal Julia script here.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"[1] Schäfer, B., Witthaut, D., Timme, M., & Latora, V. (2018). Dynamically induced cascading failures in power grids. Nature communications, 9(1), 1-13. https://www.nature.com/articles/s41467-018-04287-5","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"The system is modeled using swing equation and active power edges. The nodes are characterized by the voltage angle δ, the active power on each line is symmetric and a function of the difference between source and destination angle δ_src - δ_dst.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"using NetworkDynamics\nusing Graphs\nusing OrdinaryDiffEqTsit5\nusing DiffEqCallbacks\nusing Plots\nusing Test #hide\nimport SymbolicIndexingInterface as SII","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"For the nodes we define the swing equation. State v[1] = δ, v[2] = ω. The swing equation has three parameters: p = (P_ref, I, γ) where P_ref is the power setpopint, I is the inertia and γ is the droop or damping coeficcient.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"The output of the node is just the first state. g=1 is a shorthand for g=StateMask(1:1) which implements a trivial output function g which just takes the first element of the state vector.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"function swing_equation(dv, v, esum, p,t)\n P, I, γ = p\n dv[1] = v[2]\n dv[2] = P - γ * v[2] .+ esum[1]\n dv[2] = dv[2] / I\n nothing\nend\nvertex = VertexModel(f=swing_equation, g=1, sym=[:δ, :ω], psym=[:P_ref, :I=>1, :γ=>0.1])","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"Lets define a simple purely active power line whose active power flow is completlye determined by the connected voltage angles and the coupling constant K. We give an additonal parameter, the line limit, which we'll use later in the callback.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"function simple_edge(e, v_s, v_d, (K,), t)\n e[1] = K * sin(v_s[1] - v_d[1])\nend\nedge = EdgeModel(;g=AntiSymmetric(simple_edge), outsym=:P, psym=[:K=>1.63, :limit=>1])","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"With the definition of the graph topology we can build the Network object:","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"g = SimpleGraph([0 1 1 0 1;\n 1 0 1 1 0;\n 1 1 0 1 0;\n 0 1 1 0 1;\n 1 0 0 1 0])\nswing_network = Network(g, vertex, edge)","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"For the parameters, we create the NWParameter object prefilled with default p values","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"p = NWParameter(swing_network)\n# vertices 1, 3 and 4 act as loads\np.v[(1,3,4), :P_ref] .= -1\n# vertices 2 and 5 act as generators\np.v[(2,5), :P_ref] .= 1.5\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"We can use find_fixpoint to find a valid initial condition of the network","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"u0 = find_fixpoint(swing_network, p)\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"In order to implement the line failures, we need to create a VectorContinousCallback. In the callback, we compare the current flow on the line with the limit. If the limit is reached, the coupling K is set to 0.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"First we can define the affect function:","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"function affect!(integrator, idx)\n println(\"Line $idx tripped at t=$(integrator.t)\")\n p = NWParameter(integrator) # get indexable parameter object\n p.e[idx, :K] = 0\n auto_dt_reset!(integrator)\n save_parameters!(integrator)\n nothing\nend\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"There is one important aspect to this function: the save_parameters! call. In the callback, we change the parameters of the network, making the parameters time dependent. The flow on the line is a function P(t) = f(u(t), p(t)). Thus we need to inform the integrator, that a discrete change in parameters happend. With this, the solution object not only tracks u(t) but also p(t) and we may extract the observable P(t) directly.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"The callback trigger condition is a bit more complicated. The straight forward version looks like this:","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"function naive_condition(out, u, t, integrator)\n # careful, u != integrator.u\n # therefore construct nwstate with Network info from integrator but u\n s = NWState(integrator, u, integrator.p, t)\n for i in eachindex(out)\n out[i] = abs(s.e[i,:P]) - s.p.e[1,:limit] # compare flow with limit for line\n end\n nothing\nend\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"However, from a performacne perspectiv there are problems with this solution: on every call, we need to perform symbolic indexing into the NWState object. Symbolic indexing is not cheap, as it requires to gather meta data about the network. Luckily, the SymbolicIndexingInterface package which powers the symbolic indexing provides the lower level functions getp and getu which can be used to create and cache accessors to the internal states.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"This still isn't ideal beacuse both getlim and getflow getters will create arrays within the callback. But is far better then resolving the flat state indices every time.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"condition = let getlim = SII.getp(swing_network, epidxs(swing_network, :, :limit)),\n getflow = SII.getu(swing_network, eidxs(swing_network, :, :P))\n function (out, u, t, integrator)\n # careful, u != integrator.u\n # therefore construct nwstate with Network info from integrator but u\n s = NWState(integrator, u, integrator.p, t)\n out .= getlim(s) .- abs.(getflow(s))\n nothing\n end\nend\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"We can combine affect and condition to form the callback.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"trip_cb = VectorContinuousCallback(condition, affect!, ne(g));\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"However, there is another component missing. If we look at the powerflow on the lines in the initial steady state","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"u0.e[:, :P]","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"We see that every flow is below the trip value 1.0. Therefor we need to add a distrubance to the network. We do this by manually disabeling line 5 at time 1.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"trip_first_cb = PresetTimeCallback(1.0, integrator->affect!(integrator, 5));\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"With those components, we can create the problem and solve it.","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"prob = ODEProblem(swing_network, uflat(u0), (0,6), copy(pflat(p));\n callback=CallbackSet(trip_cb, trip_first_cb))\nsol = solve(prob, Tsit5());\n# we want to test the reconstruction of the observables # hide\n@test all(!iszero, sol(sol.t; idxs=eidxs(sol,:,:P))[begin]) # hide\n@test all(iszero, sol(sol.t; idxs=eidxs(sol,:,:P))[end][[1:5...,7]]) # hide\nnothing #hide","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"Through the magic of symbolic indexing we can plot the power flows on all lines:","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"plot(sol; idxs=eidxs(sol,:,:P))","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"","category":"page"},{"location":"generated/cascading_failure/","page":"Cascading Failure","title":"Cascading Failure","text":"This page was generated using Literate.jl.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"EditURL = \"../../examples/gas_network.jl\"","category":"page"},{"location":"generated/gas_network/#Dynamic-Flow-in-simple-Gas-Network","page":"Gas Network","title":"Dynamic Flow in simple Gas Network","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"This Example is based on the paper","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Albertus J. Malan, Lukas Rausche, Felix Strehle, Sören Hohmann, Port-Hamiltonian Modelling for Analysis and Control of Gas Networks, IFAC-PapersOnLine, Volume 56, Issue 2, 2023, https://doi.org/10.1016/j.ifacol.2023.10.193.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"and tries replicate a simple simulation of flow in a 3-node gas network.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"This example can be dowloaded as a normal Julia script here.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"We start by importing the necessary packages:","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"using NetworkDynamics\nusing ModelingToolkit\nusing DynamicQuantities\nusing ModelingToolkit: D as Dt, t as t\nusing Test\nusing StaticArrays\nusing LinearInterpolations\nusing OrdinaryDiffEqTsit5\nusing CairoMakie\nCairoMakie.activate!(type=\"svg\") #hide","category":"page"},{"location":"generated/gas_network/#Node-Models","page":"Gas Network","title":"Node Models","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"In this example, we use the equation based modeling using ModelingToolkit.jl. To verify the equations on a basic level we also provide units to eveything to perform dimensionality checks.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"There are 2 node models used in the paper. The first node type has a constant pressure. Additionally, we ad some \"internal\" state q̃_inj which we want to plot later (see also Observables).","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@mtkmodel ConstantPressureNode begin\n @parameters begin\n p_set, [description=\"Constant pressure setpoint\", unit=u\"Pa\"]\n end\n @variables begin\n p(t) = p_set, [description=\"Pressure\", unit=u\"Pa\", output=true]\n q̃_nw(t), [description=\"aggregated flow from pipes into node\", unit=u\"m^3/s\", input=true]\n q̃_inj(t), [description=\"internal state for introspection\", unit=u\"m^3/s\"]\n end\n @equations begin\n p ~ p_set\n q̃_inj ~ -q̃_nw\n end\nend\nnothing #hide","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The second node model is a variable pressure node. It has one output state (the pressure) and one input state, the aggregated flows from the connected pipes. As an internal state we have the injected flow from our source/load. The source/load behaviour itself is provided via a time dependent function.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@mtkmodel VariablePressureNode begin\n @structural_parameters begin\n load_profile # time dependent load profile\n end\n @parameters begin\n C, [description=\"Lumped capacitance of connected pipes\", unit=u\"m^4 * s^2 / kg\"]\n end\n @variables begin\n p(t)=5e6, [description=\"Pressure\", unit=u\"Pa\", output=true]\n q̃_inj(t), [description=\"external injection into node\", unit=u\"m^3/s\"]\n q̃_nw(t), [description=\"aggregated flow from pipes into node\", unit=u\"m^3/s\", input=true]\n end\n @equations begin\n q̃_inj ~ load_profile(t)\n C * Dt(p) ~ q̃_inj + q̃_nw # (30)\n end\nend\nnothing #hide","category":"page"},{"location":"generated/gas_network/#Pipe-Model","page":"Gas Network","title":"Pipe Model","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The pipe is modeld as a first order ODE for the volumetric flow at the dst end. It has two inputs: the pressure at the source and and the pressure at the destination end. Later on, we'll specify the model to be antisymmetric, thus the flow is calculated explicitly for the destination end, but the source end will just recive just that times (-1).","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@mtkmodel Pipe begin\n @parameters begin\n L, [description=\"Length of pipe\", unit=u\"m\"]\n D, [description=\"Diameter of pipe\", unit=u\"m\"]\n A, [description=\"Cross-sectional area of pipe\", unit=u\"m^2\"]\n sinθ, [description=\"Angle of inclination\" ]\n γ, [description=\"Friction efficiency factor\"]\n η, [description=\"Dynamic viscosity\", unit=u\"kg/(m*s)\"]\n r, [description=\"Pipe roughness\", unit=u\"m\"]\n g, [description=\"Gravitational acceleration\", unit=u\"m/s^2\"]\n T, [description=\"simulation temperature\", unit=u\"K\"]\n Tc, [description=\"crictical temperature\", unit=u\"K\"]\n pc, [description=\"critical pressure\", unit=us\"Pa\"]\n Rs, [description=\"Specific gas constant for natural gas\", unit=us\"J/(kg*K)\"]\n c̃, [description=\"Speed of sound in fluid at standard conditions\", unit=u\"m/s\"]\n ρ̃, [description=\"standard density\", unit=u\"kg/m^3\"]\n p̃, [description=\"standard pressure\", unit=us\"Pa\"]\n end\n @variables begin\n p_src(t), [description=\"Pressure at source end\", unit=us\"Pa\", input=true]\n p_dst(t), [description=\"Pressure at destination end\", unit=us\"Pa\", input=true]\n q̃(t)=1, [description=\"Flow through pipe\", unit=u\"m^3/s\", output=true]\n Re(t), [description=\"Reynolds number\"]\n λ(t), [description=\"Friction factor\"]\n λe(t), [description=\"Effective friction factor\"]\n pM(t), [description=\"mean pressure\", unit=us\"Pa\"]\n Z(t), [description=\"compressibility factor\"]\n ρ(t), [description=\"density\", unit=u\"kg/m^3\"]\n c(t), [description=\"speed of sound\", unit=u\"m/s\"]\n end\n @equations begin\n Z ~ 1 - 3.52 * pM/pc * exp(-2.26*(T/Tc)) + 0.274 * (pM/pc)^2 * exp(-1.878*(T/Tc)) # (5)\n ρ ~ pM / (Rs * T * Z) # (4)\n\n # TODO: Whats the correct speed of sound?\n c ~ sqrt(T * Rs * Z) # (4) # pressure/temp dependent\n # c ~ c̃ # \"standard\" speed of sound based on standard conditions\n\n # TODO: Whats the correct Reynolds number?\n Re ~ (ρ * abs(q̃*p̃/pM) * D) / (η * A) # (6) # based \"actual\" conditions\n # Re ~ (ρ̃ * abs(q̃) * D) / (η * A) # (6) # based on standard conditions\n\n λ ~ ifelse(Re < 2300,\n 64/Re, # laminar (7)\n (2*log10(4.518/Re * log10(Re/7) + r/(3.71*D)))^(-2) # turbulent (8)\n )\n λe ~ λ/γ^2 # (10)\n pM ~ 2/3*(p_src + p_dst - (p_src*p_dst)/(p_src + p_dst)) # (20)\n\n Dt(q̃) ~ A/(L*ρ̃)*(-(λe * ρ̃^2 * c^2 * L * abs(q̃))/(2 * D * A^2 * pM) * q̃ - (g * L * sinθ)/(c^2) * pM + (p_src - p_dst)) # (31)\n end\nend\nnothing #hide","category":"page"},{"location":"generated/gas_network/#Parametrization","page":"Gas Network","title":"Parametrization","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The parameterization turned out to be a bit tricky. There might be errors in there.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Some of them are quite cleare and explicitly given.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"g = 9.81u\"m/s^2\" # that we just know\nRs = 518.28u\"J/(kg*K)\" # Specific gas constant for natural gas\nη = 1e-5u\"kg/(m*s)\" # Dynamic viscosity\npc = 46.5u\"bar\" # Critical pressure\np̃ = 1.01325u\"bar\" # standard pressure\nTc = 190.55u\"K\" # critical temperature\nT̃ = 273.15u\"K\" # standard temperature\nT = 278u\"K\" # simulation temperature\nγ = 0.98 # friction efficiency factor\nr = 0.012u\"mm\" # pipe roughness\nD = 0.6u\"m\" # pipe diameter\n\n# TODO: here is switched the lenths l12 and l13. The results are better. Is this a mistake in the paper?\nL₁₂ = 90u\"km\"\nL₁₃ = 80u\"km\"\nL₂₃ = 100u\"km\"\nΔh₁ = 0u\"km\" # this value is different for different sims in the paper\np₁_set = 50u\"bar\"\nnothing # hide","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The geometric parameters for the pipes can be directly derived.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"A = π/4 * D^2\nsinθ₁₂ = ustrip(Δh₁ / L₁₂)\nsinθ₁₃ = ustrip(Δh₁ / L₁₃)\nsinθ₂₃ = 0.0\nnothing # hide","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Lastly, we need to calculate the compressibility factor, the speed of sound and the density at standard conditions:","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Z̃ = 1 - 3.52 * p̃/pc * exp(-2.26*(T̃/Tc)) + 0.274 * (p̃/pc)^2 * exp(-1.878*(T̃/Tc)) # (5)\nc̃ = sqrt(T̃ * Rs * Z̃) # (4) at standard conditions\nρ̃ = p̃ / (Rs * T̃ * Z̃) # (4) at standard conditions\n\nnothing # hide","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The equivalent \"pressure capacity\" at the nodes is calculated as a sum of the connected pipe parameters according to (28).","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Here use defintions based on the speed and \"standard\" conditions.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"C₂ = L₁₂*A/(2*ρ̃*c̃^2) + L₂₃*A/(2*ρ̃*c̃^2) # (28)\nC₃ = L₁₃*A/(2*ρ̃*c̃^2) + L₂₃*A/(2*ρ̃*c̃^2) # (28)\nnothing #hide","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Alternatively, we could calculate Z2 and Z3 based on the actuel pressure and simulation temperature. Then we could calculated the speed of sound for the \"correct\" conditions at the node. It seems to have very little effect on the actual results so I kept it simple.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"nothing #hide","category":"page"},{"location":"generated/gas_network/#Load-Profile","page":"Gas Network","title":"Load Profile","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The paper specifies the load profile at two nodes. We use the package LinearInterpolations.jl to get a callable object which represents this picewise linear interpolation.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"However, this function is not Symbolics.jl compatible, so we need to stop Symbolics.jl/ModelingToolkit.jl from tracing it. To do so, we use @register_symbolic to declare it as a symbolic function which is treated as a blackbox.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Additionally, we need to tell ModelingToolkit about the units of this object. This is just used for the static unit check during construction of the model. Later one, when we generate the Julia code from the symbolic reepresentation all units will be stripped.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"note: Discontinuities in RHS\nThe picewise linear interpolated function creates discontinuities in the RHS of the system. However since we know the times exactly, we can handle this by simply giving a list of explicit tstops to the solve command, to make sure those are hit exactly.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"load2(t) = -Interpolate(SA[0, 4, 12, 20, 24]*3600, SA[20, 30, 10, 30, 20], extrapolate=LinearInterpolations.Constant(20))(t)\nload3(t) = -Interpolate(SA[0, 4, 12, 20, 24]*3600, SA[40, 50, 30, 50, 40], extrapolate=LinearInterpolations.Constant(40))(t)\n@register_symbolic load2(t)\n@register_symbolic load3(t)\nModelingToolkit.get_unit(op::typeof(load2), _) = u\"m^3/s\"\nModelingToolkit.get_unit(op::typeof(load3), _) = u\"m^3/s\"\nnothing #hide","category":"page"},{"location":"generated/gas_network/#Building-the-Network","page":"Gas Network","title":"Building the Network","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"To bild the Network we first need to define the components. This is a two step process:","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"first create the symbolic ODESystem using ModelingToolkit\nsecondly build a NetworkDynamics component model (VertexModel/EdgeModel) based on the symbolic system.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"In the first step we can use the keyword arguments to pass \"default\" values for our parameters and states. Those values will be automaticially transfered to the metadata of the component model the second step.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"The second step requires to define the interface variables, i.e. what are the \"input\" states of your component model and what are the \"output\" states. For VertexModel the input state is the aggregated flow of all connected pipes. The output state is the pressure of the node.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@named v1_mtk = ConstantPressureNode(p_set=p₁_set)\nv1 = VertexModel(v1_mtk, [:q̃_nw], [:p]; name=:v1, vidx=1)","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@named v2_mtk = VariablePressureNode(C=C₂, load_profile=load2)\nv2 = VertexModel(v2_mtk, [:q̃_nw], [:p]; name=:v2, vidx=2)","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@named v3_mtk = VariablePressureNode(C=C₃, load_profile=load3)\nv3 = VertexModel(v3_mtk, [:q̃_nw], [:p]; name=:v3, vidx=3)","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"For the edge Model we have two inputs: the pressure on both source and destination end. There is a single output state: the volumetric flow. However we also need to tell NetworkDynamics about the coupling type. In this case we use AntiSymmetric, which meas that the source end will recieve the same flow, just inverted sign.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"@named e12_mtk = Pipe(; L=L₁₂, sinθ=sinθ₁₂, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃)\n@named e13_mtk = Pipe(; L=L₁₃, sinθ=sinθ₁₃, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃)\n@named e23_mtk = Pipe(; L=L₂₃, sinθ=sinθ₂₃, D, A, γ, η, r, g, T, Tc, pc, Rs, c̃, ρ̃, p̃)\n\ne12 = EdgeModel(e12_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e12, src=1, dst=2)\ne13 = EdgeModel(e13_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e13, src=1, dst=3)\ne23 = EdgeModel(e23_mtk, [:p_src], [:p_dst], AntiSymmetric([:q̃]); name=:e23, src=2, dst=3)","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"To build the network object we just need to pass the vertices and edges to the constructor.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Note that we've used the vidx and src/dst keywords in the constructors to define for each component to which \"part\" of the network it belongs.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"This means, the constructor can automaticially construct a graph based on those informations and we don't need to pass it explicitly.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"nw = Network([v1, v2, v3], [e12, e13, e23])","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"As a result, we recive a network with 3 unique types (v2 and v3 are similar but structurally different, because both functions capure a unique loadprofile function).","category":"page"},{"location":"generated/gas_network/#Finding-a-Steady-State","page":"Gas Network","title":"Finding a Steady State","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"To simulate the systme, we first need to find a steadystate. As a \"guess\" for that we create a NWState object from the network. This will allocate flat arrays for states u and parameters p and fill them with the default values.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"uguess = NWState(nw)","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"This is not a steadystate of the system however. To find a true steadystate we want to ensure that the lhs of the system is zero. We can solve for a steady state numerically by defining a Nonlinear Rootfind problem.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"To do so, we need to wrap the Network object in a closure.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"nwwrap = (du, u, p) -> begin\n nw(du, u, p, 0)\n nothing\nend\ninitprob = NonlinearProblem(nwwrap, uflat(uguess), pflat(uguess))\ninitsol = solve(initprob)","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"We can create a new NWState object by wrapping the solution from the nonlinear problem and the original prameters in a new NWState object.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"u0 = NWState(nw, initsol.u, uguess.p)","category":"page"},{"location":"generated/gas_network/#Solving-the-ODE","page":"Gas Network","title":"Solving the ODE","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Using this as our initial state we can create the actual ODEProblem. Since the ode allways operates on flat state and aprameter arrays we use uflat and pflat to extract them.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"prob = ODEProblem(nw, uflat(u0), (0.0,24*3600), copy(pflat(u0)))\nsol = solve(prob, Tsit5(), tstops=[0,4,12,20,24]*3600)\nnothing #hide","category":"page"},{"location":"generated/gas_network/#Inspect-the-Solution","page":"Gas Network","title":"Inspect the Solution","text":"","category":"section"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Inspecting the solution is all which is left to do.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"xticks = ((0:4:24)*3600, string.(0:4:24)) # its nice to display hours\nfig = begin\n _fig = Figure()\n row = 1\n ax = Axis(_fig[row, 1]; xlabel=\"time [h]\", ylabel=\"pressure [Pa]\", title=\"Pressure at nodes\", xticks)\n xlims!(ax, sol.t[begin], sol.t[end])\n ylims!(ax, 47.9e5, 49.9e5)\n for i in 1:3\n lines!(ax, sol, idxs=vidxs(nw, i, :p); label=\"v$i\", color=Cycled(i))\n end\n axislegend(ax)\n row += 1\n\n ax = Axis(_fig[row, 1]; xlabel=\"time [h]\", ylabel=\"flow [m³/s]\", title=\"Flow through pipes\", xticks)\n xlims!(ax, sol.t[begin], sol.t[end])\n ylims!(ax, 16, 44)\n for i in 1:2\n lines!(ax, sol, idxs=eidxs(nw, i, :q̃); label=\"e$i flow\", color=Cycled(i))\n end\n axislegend(ax, position=:rb)\n row += 1\n _fig\nend","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Notably, the \"internal\" states defined in the symbolic models are not \"states\" in the sense of the ODE. For example, we captured the load profile in the q̃_inj state of the VariablePressureNode. The only dynamic state of the model however is p. Using the \"observables\" mechanism from SciML, which is implemented by NetworkDynamics, we can reconstruct those \"optimized\" states which have been removed symbolicially. Here we plot the reconstructed load profile of nodes 2 and 3. Also, we know that node 1 is infinetly stiff, acting as an infinite source of volumetric flow. We can reconstruct this flow too.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"fig = begin\n _fig = Figure()\n row = 1\n ax = Axis(_fig[row, 1]; xlabel=\"time [h]\", ylabel=\"flow [m³/s]\", title=\"Flow at nodes\", xticks)\n xlims!(ax, sol.t[begin], sol.t[end])\n lines!(ax, sol, idxs=vidxs(nw, 1, :q̃_inj); label=\"v1 compensation\", color=Cycled(1))\n for i in 2:3\n lines!(ax, sol, idxs=vidxs(nw, i, :q̃_inj); label=\"v$i load profile\", color=Cycled(i))\n end\n axislegend(ax, position=:rc)\n _fig\nend","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"Lastly we want to observe two internal states of the pipes: the Reynolds number and the mean pressure. We see, that we're purely in the turbulent flow regime.","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"fig = begin\n _fig = Figure()\n row = 1\n ax = Axis(_fig[row, 1]; xlabel=\"time [h]\", ylabel=\"Reynolds number\", title=\"Reynolds number\", xticks)\n xlims!(ax, sol.t[begin], sol.t[end])\n for i in 1:3\n lines!(ax, sol, idxs=eidxs(nw, i, :Re); label=\"e $i\", color=Cycled(i))\n end\n hlines!(ax, 2300, color=:black, linestyle=:dash, label=\"L/T transition\")\n axislegend(ax, position=:rb)\n row += 1\n\n ax = Axis(_fig[row, 1]; xlabel=\"time [h]\", ylabel=\"Mean pressure [Pa]\", title=\"Mean pressure in pipes\", xticks)\n xlims!(ax, sol.t[begin], sol.t[end])\n for i in 1:3\n lines!(ax, sol, idxs=eidxs(nw, i, :pM); label=\"e $i\", color=Cycled(i))\n end\n axislegend(ax, position=:rb)\n _fig\nend","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"","category":"page"},{"location":"generated/gas_network/","page":"Gas Network","title":"Gas Network","text":"This page was generated using Literate.jl.","category":"page"},{"location":"#NetworkDynamics","page":"General","title":"NetworkDynamics","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"A package for working with dynamical systems on complex networks. NetworkDynamics.jl provides an interface between Graphs.jl and DifferentialEquations.jl. It allows for high performant simulation of dynamic networks by describing the local dynamics on the edges and vertices of the graph.","category":"page"},{"location":"","page":"General","title":"General","text":"The behavior of a node or an edge can be described by algebraic equations, by differential algebraic equation (DAEs) in mass matrix form or by ordinary differential equations (ODE). ","category":"page"},{"location":"","page":"General","title":"General","text":"The central construction is the function Network that receives functions describing the local dynamics on the edges and nodes of a graph g as inputs, and returns a composite function compatible with the DifferentialEquations.jl calling syntax.","category":"page"},{"location":"","page":"General","title":"General","text":"nd = Network(g, vertex_dynamics, edge_dynamics)\nnd(dx, x, p, t)","category":"page"},{"location":"","page":"General","title":"General","text":"Main features:","category":"page"},{"location":"","page":"General","title":"General","text":"Clear separation of local dynamics and topology: you can easily change topology of your system or switching out dynamical components.\nHigh performance when working with heterogeneous models (which means heaving different local dynamics in different parts of your network).\nSymbolic Indexing into solutions and states: NetworkDynamics keeps track of the states of the individual subsystems.\nDifferent execution schemes: NetworkDynamics exploits the known inter-dependencies between components to auto parallelize execution, even on GPUs!\nEquation based models: use ModelingToolkit.jl to define local dynamics, use NetworkDynamics.jl to combine them into large networks!","category":"page"},{"location":"#Where-to-begin?","page":"General","title":"Where to begin?","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"Check out the Mathematical Model to understand the underlying modelling ideas of NetworkDynamics followed by the page on Network Construction to learn how to implement you own models.","category":"page"},{"location":"","page":"General","title":"General","text":"If you prefer to look at some concrete code first check out the Getting Started tutorial!","category":"page"},{"location":"#Installation","page":"General","title":"Installation","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"Installation is straightforward with Julia's package manager.","category":"page"},{"location":"","page":"General","title":"General","text":"(v1.11) pkg> add NetworkDynamics","category":"page"},{"location":"#Reproducibility","page":"General","title":"Reproducibility","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"
Direct dependencies used for this documentation:","category":"page"},{"location":"","page":"General","title":"General","text":"using Pkg # hide\nPkg.status() # hide","category":"page"},{"location":"","page":"General","title":"General","text":"
","category":"page"},{"location":"","page":"General","title":"General","text":"
Julia Version:","category":"page"},{"location":"","page":"General","title":"General","text":"using InteractiveUtils # hide\nversioninfo() # hide","category":"page"},{"location":"","page":"General","title":"General","text":"
","category":"page"},{"location":"","page":"General","title":"General","text":"
Full Manifest:","category":"page"},{"location":"","page":"General","title":"General","text":"using Pkg # hide\nPkg.status(; mode = PKGMODE_MANIFEST) # hide","category":"page"},{"location":"","page":"General","title":"General","text":"
","category":"page"},{"location":"#Funding","page":"General","title":"Funding","text":"","category":"section"},{"location":"","page":"General","title":"General","text":"Development of this project was in part funded by the German Federal Ministry for Economic Affairs and Climate Action as part of the OpPoDyn-Project (Project ID 01258425/1, 2024-2027).","category":"page"},{"location":"","page":"General","title":"General","text":"","category":"page"},{"location":"network_construction/#Network-Construction","page":"Network Construction","title":"Network Construction","text":"","category":"section"},{"location":"network_construction/#Building-a-Network","page":"Network Construction","title":"Building a Network","text":"","category":"section"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"The main type of NetworkDyanmics.jl is a Network. A network bundles various component models (edge and vertex models) together with a graph to form a callable object which represents the RHS of the overall dynamical system, see Mathematical Model.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"A Network is build by passing a graph g, vertex models vertexm and edge models edgem.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"nw = Network(g, vertexm, edgem; kwargs...)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"Two important keywords for the Network constructor are:","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"execution: Defines the ExecutionStyle of the coreloop, e.g. SequentialExecution{true}(). A execution style is a special struct which tells the backend how to parallelize for example. A list of available executions styles can be found under Execution Types in the API.\naggregator: Tells the backend how to aggregate and which aggregation function to use. Aggregation is the process of creating a single vertex input by reducing over the outputs of adjecent edges of said vertex. The aggregator contains both the function and the algorith. E.g. SequentialAggregator(+) is a sequential aggregation by summation. A list of availabe Aggregators can be found under Aggregators in the API.","category":"page"},{"location":"network_construction/#Building-VertexModels","page":"Network Construction","title":"Building VertexModels","text":"","category":"section"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"This chapter walks through the most important aspects when defining custom vertex model. For a list of all keyword arguments please check out the docstring of VertexModel. As an example, we'll construct an second order kuramoto model, because that's what we do.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"using NetworkDynamics #hide\nfunction kuramoto_f!(dv, v, esum, p, t)\n M, P, D = p\n dv[1] = v[2]\n dv[2] = (P - D*v[2] + esum[1])/M\n nothing\nend\nfunction kuramoto_g!(y, v, esum, p, t)\n y[1] = v[1]\n nothing\nend\nVertexModel(; f=kuramoto_f!, g=kuramoto_g!, dim=2, pdim=3, outdim=1)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"Those keywords are the minimum metadata we need to provide.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"However there is a problem: the vertex is classified as a FeedForward vertex, which is unnecessary. We can improve the implementation of g according to the Feed Forward Behavior section.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"function kuramoto_g_noff!(y, v, p, t)\n y[1] = v[1]\n nothing\nend\nVertexModel(; f=kuramoto_f!, g=kuramoto_g_noff!, dim=2, pdim=3, outdim=1)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"It is still annoying to explicitly write this trivial output function. You can prevent this by using StateMask. By writing","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"VertexModel(; f=kuramoto_f!, g=StateMask(1:1), dim=2, pdim=3)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"we told the vertex model, that the output is part of the states x[1:1]. This enables a few things:","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"outdim is not needed anymore, can be inferred from StateMask\noutsym is not a generic :o any more but inferred from the state symbols.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"We can be even less verbose by writing g=1:1 or just g=1.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"In a last we define better names for our states and parameters as well as assigning a position in the graph to enable the graphless network construction. Whenever you provide sym keyword the corresponding dim keyword is not neccessary anymore. We end up with a relatively short definition","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"VertexModel(; f=kuramoto_f!, g=1,\n sym=[:θ, :ω], psym=[:M=>1, :P=>0.1, :D=>0], \n insym=[:P_nw], name=:swing, vidx=1)","category":"page"},{"location":"network_construction/#Building-EdgeModels","page":"Network Construction","title":"Building EdgeModels","text":"","category":"section"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"This chapter walks through the most important aspects when defining custom edge models. For a list of all keyword arguments please check out the docstring of EdgeModel.","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"As an example edge model we want to define standard sinusoidal coupling between the vertices in our network. The full definition looks like this:","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"function edge_f!(de, e, vsrc, vdst, p, t)\n nothing\nend\nfunction edge_g!(ysrc, ydst, e, vsrc, vdst, p, t)\n ydst[1] = p[1] * sin(vsrc[1] - vdst[1])\n ysrc[1] = -ydst[1]\nend\nEdgeModel(; f=edge_f!, g=edge_g!, dim=0, pdim=1, outdim=1)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"This is a purely \"static\" edge without internal states. This means we can omit f and dim entirely. Also, we can define a variant of g without the e input","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"function edge_g_ff!(ysrc, ydst, vsrc, vdst, p, t)\n ydst[1] = p[1] * sin(vsrc[1] - vdst[1])\n ysrc[1] = -ydst[1]\nend\nEdgeModel(;g=edge_g_ff!, pdim=1, outdim=1)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"which no classifies as a PureFeedForward edge. In cases like this, where the edge is actually anti symmetric we can alternatively define a single sided output function and wrapping it in an AntiSymmetric object","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"function edge_g_s!(ydst, vsrc, vdst, p, t)\n ydst[1] = p[1] * sin(vsrc[1] - vdst[1])\nend\nEdgeModel(;g=AntiSymmetric(edge_g_ff!), pdim=1, outdim=1)","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"which can also lead to briefer output naming. Available single sided wrappers are","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"Directed (no coupling at src), \nAntiSymmetric (same coupling at src and dst),\nSymmetric (inverse coupling at dst) and\nFiducial (define separate g for both ends).","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"Once again we can add additonal data like defining a src and dst index","category":"page"},{"location":"network_construction/","page":"Network Construction","title":"Network Construction","text":"function edge_g_s!(ydst, vsrc, vdst, p, t)\n ydst[1] = p[1] * sin(vsrc[1] - vdst[1])\nend\nEdgeModel(;g=AntiSymmetric(edge_g_ff!), psym=:K=>1, outsym=:P, insym=:θ, src=1, dst=4)","category":"page"}] +} diff --git a/previews/PR184/siteinfo.js b/previews/PR184/siteinfo.js new file mode 100644 index 00000000..e40ec43e --- /dev/null +++ b/previews/PR184/siteinfo.js @@ -0,0 +1 @@ +var DOCUMENTER_CURRENT_VERSION = "previews/PR184"; diff --git a/previews/PR184/symbolic_indexing/2131439f.svg b/previews/PR184/symbolic_indexing/2131439f.svg new file mode 100644 index 00000000..66157b47 --- /dev/null +++ b/previews/PR184/symbolic_indexing/2131439f.svg @@ -0,0 +1,52 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/symbolic_indexing/74fa3e98.svg b/previews/PR184/symbolic_indexing/74fa3e98.svg new file mode 100644 index 00000000..74f689e9 --- /dev/null +++ b/previews/PR184/symbolic_indexing/74fa3e98.svg @@ -0,0 +1,58 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/symbolic_indexing/d1c7b351.svg b/previews/PR184/symbolic_indexing/d1c7b351.svg new file mode 100644 index 00000000..1e7f8d53 --- /dev/null +++ b/previews/PR184/symbolic_indexing/d1c7b351.svg @@ -0,0 +1,48 @@ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + diff --git a/previews/PR184/symbolic_indexing/index.html b/previews/PR184/symbolic_indexing/index.html new file mode 100644 index 00000000..801fb0e6 --- /dev/null +++ b/previews/PR184/symbolic_indexing/index.html @@ -0,0 +1,51 @@ + +Symbolic Indexing · NetworkDynamics
GitHub

Symbolic Indexing

Using SciML's SymblicIndexingInterface.jl, ND.jl provides lots of methods to access and change variables and Parameters.

Setup code to make following examples work
using NetworkDynamics
+using Graphs
+using OrdinaryDiffEqTsit5
+using Plots

Provide Symbol Names

When construction component models, you can pass symbolic names using the sym and psym keywords.

function _edgef!(e, v_s, v_d, (K,), t)
+    e .= K * (v_s[1] .- v_d[1])
+end
+edgef = EdgeModel(;g=AntiSymmetric(_edgef!), outsym=[:flow], psym=[:K=>1])
EdgeModel :StaticEdgeM PureFeedForward()
+ ├─   0 states:  []  
+ ├─ 1/1 outputs: src=[₋flow] dst=[flow]
+ └─   1 param:   [K=1]

Here we created a static diffusion edge with suitable variable and parameter names. Similarly, we define the diffusion vertex with symbolic names.

function _vertexf!(dv, v, esum, p, t)
+    dv[1] = esum[1]
+end
+vertexf = VertexModel(f=_vertexf!, g=1, sym=[:storage])
VertexModel :VertexM PureStateMap()
+ ├─ 1 state:  [storage]
+ └─ 1 output: [storage]

Fundamental Symblic Indices

The default types for this access are the types VIndex, EIndex, VPIndex and EPIndex. Each of those symbolic indices consists of 2 elements: a reference to the network componen and a reference to the symbol within that component. As such, VIndex(2, :x) refers to variable with symbolic name :x in vertex number 2. EPIndex(4, 2) would refer to the second parameter of the edge component for the 4th edge.

Setup code to make following examples work
g = wheel_graph(5)
+nw = Network(g, vertexf, edgef)
+s = NWState(nw)
+s.v[:,:storage] .= randn(5)
+prob = ODEProblem(nw, uflat(s), (0,2), pflat(s))
+sol = solve(prob, Tsit5())

Those fundamental indices can be used in a lot of scenarios. Most importantly you can use them to

sol(sol.t; idxs=VIndex(1, :storage))   # extract timeseries out ouf solution object
+plot(sol; idxs=[VIndex(1, :storage), VIndex(5,:storage)]) # plot storage of two nodes
Example block output

Generate Symbolic Indices

Often, you need many individual symbolic indices. For that there are the helper methods vidxs, eidxs, vpidxs and epidxs. With the help of those methods you can generate arrays of symbolic indices:

vidxs(nw, :, :storage) # get variable "storage" for all nodes
5-element Vector{VIndex}:
+ VIndex(1, :storage)
+ VIndex(2, :storage)
+ VIndex(3, :storage)
+ VIndex(4, :storage)
+ VIndex(5, :storage)
plot(sol; idxs=vidxs(nw, :, :storage))
Example block output

NWState and NWParameter Objects

Internally, both state and parameters of a Network are represented using flat arrays. To access the state or parameters of a network, you can use the NWState and NWParameter objects.

p = NWParameter(nw)
Parameter{Vector{Float64}} of Network (5 vertices, 8 edges)
+  ├─ EPIndex(1, :K) => 1.0
+  ├─ EPIndex(2, :K) => 1.0
+  ├─ EPIndex(3, :K) => 1.0
+  ├─ EPIndex(4, :K) => 1.0
+  ├─ EPIndex(5, :K) => 1.0
+  ├─ EPIndex(6, :K) => 1.0
+  ├─ EPIndex(7, :K) => 1.0
+  └─ EPIndex(8, :K) => 1.0

creates a NWParameter object for the network nw. It essentially creates a new flat parameter array and fills it with the default parameter values define in the component. The parameters in the NWParameter object can be accessed using the symbolic indices.

p[EPIndex(5, :K)] = 2.0 # change the parameter K of the 5th edge

Similarly, you can create a NWState object for the network nw using

s = NWState(nw)
NWState{Vector{Float64}} of Network (5 vertices, 8 edges)
+  ├─ VIndex(1, :storage) => NaN
+  ├─ VIndex(2, :storage) => NaN
+  ├─ VIndex(3, :storage) => NaN
+  ├─ VIndex(4, :storage) => NaN
+  └─ VIndex(5, :storage) => NaN
+ p = NWParameter([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
+ t = nothing

No default values were provided in the network components, so the state array is filled with NaNs.

s[VIndex(:, :storage)] .= randn(5) # set the (initial) storage for alle nodes
NWState{Vector{Float64}} of Network (5 vertices, 8 edges)
+  ├─ VIndex(1, :storage) => 0.3953170217786552
+  ├─ VIndex(2, :storage) => 0.9173896583682629
+  ├─ VIndex(3, :storage) => -1.0978639190905948
+  ├─ VIndex(4, :storage) => 2.2930233712677026
+  └─ VIndex(5, :storage) => 1.8225484084116617
+ p = NWParameter([1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0])
+ t = nothing

For both NWState and NWParameter objects, the there is a more convenient way to access the variables and parameters.

@assert s.v[1, :storage] == s[VIndex(1, :storage)] # s.v -> access vertex states
+@assert s.e[1, :flow]    == s[EIndex(1, :flow)]    # s.e -> access edge states
+@assert s.p.e[1,:K]      == p[EPIndex(1, :K)]      # s.p -> access parameters

The NWState and NWParameter objects are mutable, thus changing them will also change the underlying wrapped flat arrays. You can allways access the flat representations by calling uflat and pflat.

Note

The NWState and NWParameter wrappers can be constructed from various objects. Fore example, within a callback you might construct p = NWParameter(integrator) to then change the parameters of the network within the callback.

Observables

Sometimes, the "states" you're interested in aren't really states in the DAE sense but rather algebraic derivations from DAE states, parameters and time – in accordance with the naming in the SciML-ecosystem those states are called Observables.

A prime example of Observables are edge/vertex-outputs, such as the flow in the edge model defined above. It is also possible to define additional Observables manually by using the obssym and obsf keyword on the EdgeModel/VertexModel constructors. When building models using ModelingToolkit, the reduced algebraic states will be preserved as observables automatically.

Observables can be accessed like any other state, for example, the flows in the network don't show up in the state array but can be accessed in all the ways discussed above, for example

plot(sol; idxs=eidxs(nw, :, :flow))
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