Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs

TL;DR

This paper establishes rigorous bounds on the generalization error of Physics Informed Neural Networks (PINNs) when used to approximate solutions to PDEs, linking training error, sample size, and stability properties.

Contribution

It introduces an abstract formalism to derive generalization error bounds for PINNs, supported by theoretical analysis and numerical validation on nonlinear PDEs.

Findings

Derived upper bounds on PINNs' generalization error

Linked error bounds to training error and sample size

Validated theory with numerical experiments on nonlinear PDEs

Abstract

Physics informed neural networks (PINNs) have recently been widely used for robust and accurate approximation of PDEs. We provide rigorous upper bounds on the generalization error of PINNs approximating solutions of the forward problem for PDEs. An abstract formalism is introduced and stability properties of the underlying PDE are leveraged to derive an estimate for the generalization error in terms of the training error and number of training samples. This abstract framework is illustrated with several examples of nonlinear PDEs. Numerical experiments, validating the proposed theory, are also presented.