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

TL;DR

This paper provides rigorous theoretical estimates on the generalization error of Physics Informed Neural Networks (PINNs) when used for inverse problems related to PDEs, supported by numerical validation.

Contribution

It introduces an abstract framework and derives error estimates for PINNs in inverse PDE problems, specifically for data assimilation and unique continuation tasks.

Findings

Rigorous generalization error bounds for PINNs in inverse PDE problems.

Validation of theoretical estimates through numerical experiments.

Application to four prototypical linear PDEs.

Abstract

Physics informed neural networks (PINNs) have recently been very successfully applied for efficiently approximating inverse problems for PDEs. We focus on a particular class of inverse problems, the so-called data assimilation or unique continuation problems, and prove rigorous estimates on the generalization error of PINNs approximating them. An abstract framework is presented and conditional stability estimates for the underlying inverse problem are employed to derive the estimate on the PINN generalization error, providing rigorous justification for the use of PINNs in this context. The abstract framework is illustrated with examples of four prototypical linear PDEs. Numerical experiments, validating the proposed theory, are also presented.