Q19Questions

What is a physics-informed neural network?

Short answer

A physics-informed neural network (PINN) is a neural network trained not only on data but also on the equations of physics: how far its prediction violates the governing equations, such as conservation of mass, momentum or energy, is part of its loss function. This lets it learn from little data and keeps its predictions physically consistent. The approach was popularised by Raissi, Perdikaris and Karniadakis in 2017–2019.

How it works

A normal network learns only from examples. A PINN is also penalised whenever its output breaks a physical law. Because neural networks can be differentiated automatically, the residual of a differential equation, for example the heat equation, can be computed at any point and added to the training loss.

Neural operators

A related idea learns a whole solution operator instead of one solution: the Fourier neural operator maps a shape and its boundary conditions to the full flow or temperature field. Once trained, it evaluates new cases in milliseconds, which makes large design searches affordable.

Limits

PINNs can be hard to train on problems with sharp gradients, such as shocks, or with very different scales at once. In engineering they are best used together with conventional solvers: trained and checked against them, not instead of them.

Sources

Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equationsMaziar Raissi, Paris Perdikaris, George E. Karniadakis · 2019 · Journal of Computational Physics 378, 686–707
Fourier neural operator for parametric partial differential equationsZongyi Li et al. · 2020 · arXiv 2010.08895

Last updated 2026-10-21