Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

TL;DR

This paper develops a theoretical and computational framework for certifying the prediction error of physics-informed neural networks (PINNs), extending its applicability to complex problems like Stokes flow, with demonstrated practical success.

Contribution

It extends a semigroup-based error estimation method for PINNs to broader problems by modifying bounds and introducing numerical strategies, enabling certification in realistic scenarios.

Findings

Extended error bounds for PINNs applicable to complex PDEs.

Numerical strategies for approximating stability parameters.

Successful certification of PINN predictions in Stokes flow.

Abstract

Rigorous error estimation is a fundamental topic in numerical analysis. With the increasing use of physics-informed neural networks (PINNs) for solving partial differential equations, several approaches have been developed to quantify the associated prediction error. In this work, we build upon a semigroup-based framework previously introduced by the authors for estimating the PINN error. While this estimator has so far been limited to academic examples - due to the need to compute quantities related to input-to-state stability - we extend its applicability to a significantly broader class of problems. This is accomplished by modifying the error bound and proposing numerical strategies to approximate the required stability parameters. The extended framework enables the certification of PINN predictions in more realistic scenarios, as demonstrated by a numerical study of Stokes flow around a cylinder.