Evaluating Error Bound for Physics-Informed Neural Networks on Linear Dynamical Systems

TL;DR

This paper derives explicit analytical error bounds for physics-informed neural networks applied to linear differential systems, linking residuals to solution accuracy and validating bounds through empirical tests.

Contribution

It introduces a method to compute explicit error bounds for PINNs on linear systems, independent of network architecture or solution knowledge.

Findings

Error bounds are strictly valid for linear systems.

Residual evaluation effectively estimates solution error.

Empirical verification confirms bounds are reliable.

Abstract

There have been extensive studies on solving differential equations using physics-informed neural networks. While this method has proven advantageous in many cases, a major criticism lies in its lack of analytical error bounds. Therefore, it is less credible than its traditional counterparts, such as the finite difference method. This paper shows that one can mathematically derive explicit error bounds for physics-informed neural networks trained on a class of linear systems of differential equations. More importantly, evaluating such error bounds only requires evaluating the differential equation residual infinity norm over the domain of interest. Our work shows a link between network residuals, which is known and used as loss function, and the absolute error of solution, which is generally unknown. Our approach is semi-phenomonological and independent of knowledge of the actual solution or the complexity or architecture of the network. Using the method of manufactured solution on linear ODEs and system of linear ODEs, we empirically verify the error evaluation algorithm and demonstrate that the actual error strictly lies within our derived bound.