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dp_solver.py
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dp_solver.py
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from jax import config
config.update("jax_enable_x64", True)
import numpy as np
import jax
import jax.numpy as jnp
from utils import plot_heatmap, plot_ref_and_test_heatmap, plot_ref_initial_and_test_heatmap
from functools import partial
import optax
Nx = 200
Lx = 2
Nt = 200
Lt = 1
dt = Lt/(Nt-1)
dx = Lx/(Nx-1)
diffusivity = 0.5
r = dt/(2*dx**2)*diffusivity
optimstep = 500
learning_rate = 0.005
#transition_steps = 500
#decay_rate = 0.2
number_sensors = 20
W_and_B_Tracking = False
if W_and_B_Tracking== True:
import wandb
wandb.init(project="InverseHeat2")
wandb.config = {"epochs": optimstep, "learning_rate": learning_rate,
#"transition_steps":transition_steps, "decay_rate": decay_rate,
"number_sensors":number_sensors}
from jax import random
key = random.PRNGKey(0)
x = np.linspace(-Lx/2,Lx/2,Nx)
t = np.linspace(0,Lt,Nt)
T_0_function = lambda x: 0*x+1
T_0 = T_0_function(x)
test_case = "Gaussian"
if test_case == "Constant":
ref_source_function = lambda x,t : 3+0*x
elif test_case == "Piecewise":
ref_source_function = lambda x,t: 1. if(x>-0.3 and x<0.3) else 0.
elif test_case == "Gaussian":
ref_source_function = lambda x,t: np.exp(-(x-0.25*np.sin(t*4))**2/0.3)
else:
raise(ValueError("Please input a valid test case name"))
meshgrid = np.meshgrid(x, t)
ref_source = np.vectorize(ref_source_function)(meshgrid[0],meshgrid[1])
initial_source= jnp.ones_like(ref_source)
source = initial_source
sensor_positions = np.linspace(x[0],x[-1],number_sensors)
def find_sensor_indices(x,sensor_positions):
sensor_indices = []
for sensor_position in sensor_positions:
sensor_indices.append(np.argmin(np.abs(x-sensor_position)))
return np.array(sensor_indices)
sensor_ind = find_sensor_indices(x,sensor_positions)
sensor_positions = x[sensor_ind]
def create_2nd_der_matrix(Nx):
central_diff_matrix = (np.diag(np.ones(Nx-1),1) - 2*np.diag(np.ones(Nx),0)+ np.diag(np.ones(Nx-1),-1))
#Incorporate constant boundary conditions by setting the first and last row to zero
central_diff_matrix[0,:]= 0
central_diff_matrix[-1,:]= 0
return central_diff_matrix
def CN2_lhs_rhs_creation(Nx):
central_diff_matrix = create_2nd_der_matrix(Nx)
lhs = np.eye(Nx) - r*(central_diff_matrix)
rhs = np.eye(Nx) + r*(central_diff_matrix)
return lhs, rhs
def create_step_fn():
lhs, rhs = CN2_lhs_rhs_creation(Nx)
def step_fn(temperature,source):
temperature = jax.scipy.linalg.solve(lhs,rhs@temperature+dt*source)
return temperature
return step_fn
def create_rollout_func():
step_fn = create_step_fn()
def scan_fn(T,source):
T_next = step_fn(T,source)
return T_next, T_next
def rollout_fn(source):
_, trj = jax.lax.scan(scan_fn,T_0,source,Nt)
return trj
return rollout_fn
def create_sensor_rollout_func():
step_fn = create_step_fn()
def scan_fn(T,source):
T_next = step_fn(T,source)
return T_next, T_next[sensor_ind]
def rollout_fn(source):
_, trj = jax.lax.scan(scan_fn,T_0,source,Nt)
return trj
return jax.jit(rollout_fn)
rollout_fn = create_rollout_func()
sensor_rollout_fn = create_sensor_rollout_func()
ref_trj = rollout_fn(source=ref_source)
plot_heatmap(ref_trj,Lt,Lx)
ref_sensor_trj = sensor_rollout_fn(ref_source)
def loss_fun(params,ref_sensor_trj):
curr_test_sensor_trj = sensor_rollout_fn(params[0])
loss = 0.
loss += jnp.linalg.norm(ref_sensor_trj-curr_test_sensor_trj)
return loss
#schedule = optax.exponential_decay(learning_rate, transition_steps= transition_steps, decay_rate=decay_rate, staircase=True)
optimizer = optax.adam(learning_rate=learning_rate)
initial_params = [source]
params = initial_params
opt_state = optimizer.init(params)
for i in range(optimstep):
loss,grad = jax.value_and_grad(loss_fun)(params,ref_sensor_trj)
print("Optimization step: ", i, "loss: ", loss)
updates, opt_state = optimizer.update(grad, opt_state)
params = optax.apply_updates(params, updates)
initial_test_trj = rollout_fn(initial_source)
final_test_trj = rollout_fn(source=params[0])
temperature_plot = plot_ref_and_test_heatmap(ref_trj,final_test_trj,Lt,Lx,x,test_case)
source_plot = plot_ref_and_test_heatmap(ref_source,params[0],Lt,Lx,x,test_case)
source_plot.savefig(f"./figures/{test_case}_source.png")
temperature_plot.savefig(f"./figures/{test_case}_temperature.png")
initial_source_loss = jnp.linalg.norm(initial_source-ref_source)
final_source_loss = jnp.linalg.norm(params[0]-ref_source)
initial_temperature_loss = jnp.linalg.norm(initial_test_trj-ref_trj)
final_temperature_loss = jnp.linalg.norm(final_test_trj - ref_trj)
print(f"Loss for source. Epoch 0: {initial_source_loss}, Final Epoch {final_source_loss}")
print(f"Loss for temperature. Epoch 0: {initial_temperature_loss}, Final Epoch {final_temperature_loss}")