Exact and approximate error bounds for physics-informed neural networks

TL;DR

This paper develops methods to calculate both exact and approximate error bounds for physics-informed neural networks solving nonlinear first-order ODEs, enabling better understanding of solution accuracy without numerical solutions.

Contribution

It introduces a general error expression for PINNs on nonlinear first-order ODEs and proposes techniques for exact and approximate error bounds based solely on residuals and equation structure.

Findings

Error bounds can be computed without numerical solutions.

The methods successfully provide bounds for specific cases.

Approximate bounds are feasible for general cases.

Abstract

The use of neural networks to solve differential equations, as an alternative to traditional numerical solvers, has increased recently. However, error bounds for the obtained solutions have only been developed for certain equations. In this work, we report important progress in calculating error bounds of physics-informed neural networks (PINNs) solutions of nonlinear first-order ODEs. We give a general expression that describes the error of the solution that the PINN-based method provides for a nonlinear first-order ODE. In addition, we propose a technique to calculate an approximate bound for the general case and an exact bound for a particular case. The error bounds are computed using only the residual information and the equation structure. We apply the proposed methods to particular cases and show that they can successfully provide error bounds without relying on the numerical solution.