Robust error estimates of PINN in one-dimensional boundary value problems for linear elliptic equations

TL;DR

This paper provides rigorous error estimates for physics-informed neural networks (PINN) applied to one-dimensional linear elliptic boundary value problems, demonstrating robustness against coefficient variations.

Contribution

It establishes existence, uniqueness, and error bounds for PINN solutions using Sobolev space theory and variational methods, independent of coefficient magnitudes.

Findings

Error is dominated by training loss.

Error-to-loss ratio decreases with larger coefficients.

Results confirm robustness of PINN error estimates.

Abstract

In this paper, we study physics-informed neural networks (PINN) to approximate solutions to one-dimensional boundary value problems for linear elliptic equations and establish robust error estimates of PINN regardless of the quantities of the coefficients. In particular, we rigorously demonstrate the existence and uniqueness of solutions using the Sobolev space theory based on a variational approach. Deriving $L^2$-contraction estimates, we show that the error, defined as the mean square of the differences between the true solution and our trial function at the sample points, is dominated by the training loss. Furthermore, we show that as the quantities of the coefficients for the differential equation increase, the error-to-loss ratio rapidly decreases. Our theoretical and experimental results confirm the robustness of the error regardless of the quantities of the coefficients.