Error Estimation for Physics-informed Neural Networks Approximating Semilinear Wave Equations

TL;DR

This paper derives rigorous error bounds for physics-informed neural networks approximating semilinear wave equations, linking total error to network size, training points, and training error, supported by numerical experiments.

Contribution

It provides the first rigorous error bounds for PINNs applied to semilinear wave equations, connecting approximation errors to network architecture and training data.

Findings

Total error can be made arbitrarily small under certain assumptions.

Error bounds relate network width and training points to approximation accuracy.

Numerical experiments validate the theoretical error estimates.

Abstract

This paper provides rigorous error bounds for physics-informed neural networks approximating the semilinear wave equation. We provide bounds for the generalization and training error in terms of the width of the network's layers and the number of training points for a tanh neural network with two hidden layers. Our main result is a bound of the total error in the $H^1([0,T];L^2(\Omega))$-norm in terms of the training error and the number of training points, which can be made arbitrarily small under some assumptions. We illustrate our theoretical bounds with numerical experiments.