Exact and approximate error bounds for physics-informed neural networks

TL;DR

This work reports important progress in calculating error bounds of physics-informed neural networks solutions of nonlinear first-order ODEs and proposes a technique to calculate an approximate bound for the general case and an exact bound for a particular case.

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.

Figures (1)

• Figure 1: The absolute value of the error of PINNs with progressively lower loss, $L$, against approximate bounds for Eq. \ref{['eq:pop']} (a) and exact bounds for Eq. \ref{['eq:cosmo']} (b).