jax-fenics ·

This package enables use of FEniCS for solving differentiable variational problems in JAX.

Automatic tangent linear and adjoint solvers are implemented for FEniCS programs involving fenics.solve and fenics.assemble. These solvers make it possible to use JAX's forward and reverse Automatic Differentiation with FEniCS.

Current limitations:

• Composition of forward and reverse modes for higher-order derivatives is not implemented yet.
• Differentiating through time-dependent FEniCS programs is not supported, for this case check out jax-fenics-adjoint or jax-firedrake.
  • It is possible to write differentiable time-stepping in JAX, check the Burgers' equation example.

Example

Here is the demonstration of solving the Poisson's PDE on 2D square domain and calculating the solution Jacobian matrix (du/df) using the reverse (adjoint) mode Automatic Differentiation.

import jax
import jax.numpy as np
from jax.config import config
config.update("jax_enable_x64", True)

import fenics
import ufl

from jaxfenics import build_jax_solve_eval
from jaxfenics import numpy_to_fenics

# Create mesh for the unit square domain
n = 10
mesh = fenics.UnitSquareMesh(n, n)

# Define discrete function spaces and functions
V = fenics.FunctionSpace(mesh, "CG", 1)
W = fenics.FunctionSpace(mesh, "DG", 0)

# Define FEniCS template representation of JAX input
templates = (fenics.Function(W),)

@build_jax_solve_eval(templates)
def fenics_solve(f):
    u = fenics.Function(V, name="PDE Solution")
    v = fenics.TestFunction(V)
    inner, grad, dx = ufl.inner, ufl.grad, ufl.dx
    F = (inner(grad(u), grad(v)) - f * v) * dx
    bcs = [fenics.DirichletBC(V, 0.0, "on_boundary")]
    fenics.solve(F == 0, u, bcs)
    # output should be a tuple (solution, F, bcs)
    return u, F, bcs

# build_jax_solve_eval is a wrapper decorator that registers
# `fenics_solve` for JAX and makes fenics_solve to return only the PDE solution

# Let's create a vector of ones with size equal to the number of cells in the mesh
f = np.ones(W.dim())
u = fenics_solve(f) # u is JAX's array
u_fenics = numpy_to_fenics(u, fenics.Function(V)) # we need to explicitly provide template function for conversion

# now we can calculate vector-Jacobian product with `jax.vjp`
jvp_result = jax.vjp(fenics_solve, f)[1](np.ones_like(u))

# or the full (dense) Jacobian matrix du/df with `jax.jacrev`
dudf = jax.jacrev(fenics_solve)(f)

# our function fenics_solve maps R^200 (dimension of W) to R^121 (dimension of V)
# therefore the Jacobian matrix dimension is dim V x dim W
assert dudf.shape == (V.dim(), W.dim())

Check examples/ or tests/ folders for the additional examples.

Installation

First install FEniCS. Then install JAX with:

python -m pip install --upgrade jax==0.1.61 jaxlib==0.1.42  # CPU-only version

After that install the jax-fenics with:

python -m pip install git+https://github.com/IvanYashchuk/jax-fenics.git@master

Reporting bugs

If you found a bug, create an issue.

Contributing

Pull requests are welcome from everyone.

Fork, then clone the repository:

git clone https://github.com/IvanYashchuk/jax-fenics.git

Make your change. Add tests for your change. Make the tests pass:

pytest tests/

Check the formatting with black and flake8. Push to your fork and submit a pull request.