EXPLAINER

Physics-Informed Neural Networks (PINNs)

techniquefreshness: evolvingsources not yet verified

PINNs train neural networks to satisfy the governing equations of a physical system (as differential-equation residuals) in addition to fitting data, to solve or learn from scientific problems with limited data.

Short answer: A PINN adds a physics residual loss (the PDE/ODE the system must obey) to the data loss, so the network respects known laws. Useful for forward/inverse problems with sparse data; can be hard to train for stiff or multi-scale problems, and classical solvers often win when they apply.

What is it?

Normally a neural network learns only from examples. A PINN is also told the physical law the system follows, and is penalised when its output breaks it. That lets it learn from less data and stay physically plausible.

Also known as: PINN, PINNs, physics informed machine learning, neural PDE solver, neural operators, Fourier neural operator, FNO, scientific ML surrogates.

Technical definition

Raissi, Perdikaris & Karniadakis (2019) formulate PINNs: network u_θ(x,t) is trained with a loss combining data misfit, initial/boundary conditions and the PDE residual evaluated by automatic differentiation at collocation points. Variants include conservative/hard-constrained formulations, domain decomposition and operator learning (neural operators such as Fourier Neural Operator, Li et al., 2020) that learn solution maps across parameter families. Known issues: loss-balancing, spectral bias, optimisation difficulties and lack of guarantees compared to numerical PDE solvers.

Entity type: technique · Freshness class: evolving.

When should I use it?

Consider for inverse problems, sparse-data settings, and parametrised families where fast surrogates matter.

Implementation patterns

  • Sample collocation points; weight data/physics losses; use hard constraints for boundaries.
  • Neural operators when many parameter settings must be solved.
  • Compare against a classical solver baseline.
  • Use uncertainty estimates.

Technologies and frameworks

Commonly associated: DeepXDE, NVIDIA Modulus/PhysicsNeMo (verify name), JAX, PyTorch, neuraloperator library. Names are examples of what exists, not recommendations, and projects change quickly.

Practical guide

  1. Write the equations and boundary conditions.
  2. Nondimensionalise.
  3. Implement residual losses.
  4. Benchmark vs classical solver.
  5. Validate across regimes.

Common failure modes

  • Poor convergence for stiff/multi-scale PDEs.
  • Loss terms imbalanced.
  • No error bounds.
  • Winning claims against weak baselines.

Troubleshooting

  • Training stalls or fails — Rescale variables, reweight losses, use curriculum or domain decomposition.
  • Doubt about accuracy — Compare against a trusted numerical solution on test cases.

Contrasted with

Questions this page answers

  • What is a physics-informed neural network?
  • PINN vs finite element method?
  • What are neural operators?
  • When should I use a PINN?

Typical goals: solve or learn differential equations with ML; decide between PINNs and classical solvers.

Sources and verification

Status: not independently verified. This entry was authored on 2026-10-06 from general technical knowledge, without live checking of the sources below. Treat every source as a lead to confirm (identifier, title, authorship, current version) before publication or citation.

Freshness class: evolving — Established techniques whose best practice shifts as research and tooling move. Re-review every 6 months.

Canonical sources (unverified leads):

  • Raissi, Perdikaris & Karniadakis, Physics-informed neural networks, J. Comput. Phys. 378 (2019) — doi:10.1016/j.jcp.2018.10.045
  • Li et al., Fourier Neural Operator for Parametric Partial Differential Equations (2020) — arXiv:2010.08895