Physics-Informed Neural Networks

PDE-constrained learninginductive biasautomatic differentiation

Neural networks that learn physical laws, not just data. The PDE goes into the loss function, so the network respects physics by construction.

Approximate the solution to a PDE:

$N [u] (t, x) = 0$

Train by minimizing a composite loss — data fidelity plus physics compliance:

$L = L_{data} + λ L_{physics}$

Partial derivatives come from autodiff through the network. Collocation points enforce the constraint across the domain — no mesh needed.

Soft PDE constraints — physics lives in the loss, not the architecture; works for any differentiable PDE
Symmetry-aware layers — equivariant structure enforces rotational and translational invariances directly
Inverse problems — unknown parameters (viscosity, diffusivity, reaction rates) are learned alongside the solution
Data efficiency — physics regularization cuts labeled data requirements by orders of magnitude