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Getting Started

FEAX (Finite Element Analysis with JAX) is a fully differentiable finite element engine. Every stage — from assembly to solve — runs on XLA and is compatible with jax.jit, jax.grad, and jax.vmap, enabling gradient-based optimization and machine learning directly on PDE simulations.

Why FEAX?​

  • JAX Transformations: Solvers work seamlessly with jax.jit, jax.grad, and jax.vmap, and arbitrary compositions such as jit(grad(...)).
  • GPU Direct Solver: Native cuDSS integration for sparse direct solves on GPU, with automatic matrix property detection (General / Symmetric / SPD).
  • End-to-End Differentiability: Gradients flow through assembly, boundary conditions, linear/nonlinear solvers, and post-processing — enabling topology optimization, inverse problems, and physics-informed learning.
  • Neural Network Research: The consistent differentiability makes FEAX a natural building block for coupling neural networks with PDE solvers.

Quick Example​

3D cantilever beam under traction — solve and compute gradients in a few lines:

import feax as fe
import jax
import jax.numpy as np

# Mesh and material
mesh = fe.mesh.box_mesh((100, 10, 10), mesh_size=2)
E, nu = 70e3, 0.3

# Define the constitutive law
class LinearElasticity(fe.problem.Problem):
    def get_tensor_map(self):
        def stress(u_grad, *args):
            mu = E / (2. * (1. + nu))
            lmbda = E * nu / ((1 + nu) * (1 - 2 * nu))
            eps = 0.5 * (u_grad + u_grad.T)
            return lmbda * np.trace(eps) * np.eye(self.dim) + 2 * mu * eps
        return stress

    def get_surface_maps(self):
        def surface_map(u, x, traction_mag):
            return np.array([0., 0., traction_mag])
        return [surface_map]

left  = lambda point: np.isclose(point[0], 0.,   atol=1e-5)
right = lambda point: np.isclose(point[0], 100., atol=1e-5)

problem = LinearElasticity(mesh, vec=3, dim=3, location_fns=[right])

# Boundary conditions: fix the left face
bc_config = fe.DCboundary.DirichletBCConfig([
    fe.DCboundary.DirichletBCSpec(location=left, component="all", value=0.)
])
bc = bc_config.create_bc(problem)

# Internal variables (surface traction magnitude)
traction = fe.InternalVars.create_uniform_surface_var(problem, 1e-3)
internal_vars = fe.InternalVars(volume_vars=(), surface_vars=[(traction,)])

# Create solver (auto-selects cuDSS on GPU, sparse direct on CPU)
solver = fe.create_solver(problem, bc,
    solver_options=fe.DirectSolverOptions(), iter_num=1,
    internal_vars=internal_vars)
initial = fe.zero_like_initial_guess(problem, bc)

# Solve
sol = solver(internal_vars, initial)

# Differentiate through the entire solve
grad_fn = jax.grad(lambda iv: np.sum(solver(iv, initial) ** 2))
grads = grad_fn(internal_vars)

Limitations​

  • First-order differentiation only: Solvers use custom_vjp internally, so jax.grad (first-order) is supported but jax.hessian (second-order) is not.
  • Time integration is user-driven: FEAX provides pipeline abstractions (ImplicitPipeline, ExplicitPipeline) but does not include built-in adaptive error control or multi-rate schemes.
  • Element order up to quadratic: Linear (degree 1) and quadratic (degree 2) elements are supported. Cubic or higher-order elements are not.
  • Fixed mesh: The mesh topology must remain constant throughout JAX transformations. No adaptive remeshing or h-refinement during differentiation.
  • Single machine: No MPI or distributed computing support. Parallelism is limited to JAX's device-level parallelism (vmap).

Overview​

New to FEAX? The Overview page summarizes the core workflow — Mesh → Problem → Boundary Conditions → Solver → Solution — and the key concepts you need to get started.

Installation​

pip install feax[cuda13]
pip install --no-build-isolation git+https://github.com/johnviljoen/spineax.git

See the Installation page for detailed installation instructions.

License​

FEAX is licensed under the GNU General Public License v3.0.

Acknowledgments​

FEAX builds upon the excellent work of:

  • JAX for automatic differentiation and compilation
  • JAX-FEM for inspiration and reference implementations
  • Spineax for cuDSS solver implementation