Finite Element Chain Rules ·

Easy interoperability with Automatic Differentiation libraries through NumPy interface to Firedrake and FEniCS.

Overview

This package provides a high-level interface for evaluating derivatives of FEniCS and Firedrake models. It is intended to be used as the backend for extending Automatic Differentiation (AD) libraries to support FEniCS and Firedrake solvers.

Automatic tangent linear and adjoint solvers for FEniCS/Firedrake problems are derived with dolfin-adjoint/pyadjoint. These solvers make it possible to use forward and reverse modes Automatic Differentiation with FEniCS/Firedrake.

This package is used for building bridges between FEniCS/Firedrake and:

• JAX in jax-fenics-adjoint,
• PyMC3 (Theano) in fenics-pymc3,
• Julia's ChainRules.jl in PyFenicsAD.jl.

Current limitations:

• Composition of forward and reverse modes for higher-order derivatives is not implemented yet.
• Differentiation with respect to mesh coordinates is not implemented yet.

API

The package includes 5 functions:

• two functions for converting between NumPy and FEniCS/Firedrake: to_numpy, from_numpy,
• evaluate_primal - computes the output of a FEniCS/Firedrake function and saves a corresponding computational graph,
• evaluate_pullback - propagates the derivative information from outputs to inputs (reverse-mode AD),
• evaluate_pushforward - propagates the derivative information from inputs to outputs (forward-mode AD).

Example

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

import numpy as np

import firedrake
import firedrake_adjoint
import ufl

from functools import partial

from fecr import evaluate_primal, evaluate_pullback
from fecr import from_numpy

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

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

# Define Firedrake template representation of NumPy input
templates = (firedrake.Function(W),)

# This function takes Firedrake types as arguments and returns a Firedrake Function (solution)
def firedrake_solve(f):
    # This function inside should be traceable by firedrake_adjoint
    u = firedrake.Function(V, name="PDE Solution")
    v = firedrake.TestFunction(V)
    inner, grad, dx = ufl.inner, ufl.grad, ufl.dx
    F = (inner(grad(u), grad(v)) - f * v) * dx
    bcs = [firedrake.DirichletBC(V, 0.0, "on_boundary")]
    firedrake.solve(F == 0, u, bcs)
    return u

# Let's build a decorator which transforms NumPy input to Firedrake types input
# and returns NumPy representation of Firedrake output
numpy_firedrake_solve = partial(evaluate_primal, firedrake_solve, templates)

# Let's create a vector of ones with size equal to the number of cells in the mesh
f = np.ones(W.dim())
u = numpy_firedrake_solve(f)[0] # u is a NumPy array now
u_firedrake = from_numpy(u, firedrake.Function(V)) # we need to explicitly provide template function for conversion

# Now let's evaluate the vector-Jacobian product
numpy_output, firedrake_output, firedrake_inputs, tape = numpy_firedrake_solve(f)
g = np.ones_like(numpy_output)

# `vjp_out` is the result of (implicitly) multiplying the vector `g` with the solution Jacobian du/df
vjp_out = evaluate_pullback(firedrake_output, firedrake_inputs, tape, g)

Check the tests/ folder for the additional usage examples.

Installation

First install FEniCS or Firedrake. Then install dolfin-adjoint for FEniCS with:

python -m pip install git+https://github.com/dolfin-adjoint/[email protected]

or for Firedrake with:

python -m pip install git+https://github.com/dolfin-adjoint/pyadjoint.git@master

Then install fecr with:

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

Reporting bugs

If you found a bug, create an issue.

Asking questions and general discussion

If you have a question or anything else, create a new discussion. Using issues is also fine!

Contributing

Pull requests are welcome from everyone.

Fork, then clone the repository:

git clone https://github.com/IvanYashchuk/fecr.git

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

pytest tests/firedrake_backend  # or pytest tests/fenics_backend

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