Numerical Robustness of PINNs for Multiscale Transport Equations

TL;DR

This paper examines the numerical robustness of Physics Informed Neural Networks (PINNs) for multiscale transport equations, revealing limitations in diffusive regimes and proposing a diffusive scaling to improve accuracy, supported by theoretical and numerical analysis.

Contribution

It establishes the analogy between PINNs and Least-Squares Finite Elements for multiscale transport equations and introduces a diffusive scaling to overcome identified limitations.

Findings

PINNs do not reach the correct diffusive limit without scaling.

A diffusive scaling improves the accuracy of PINNs in the diffusive regime.

Numerical results support the theoretical analysis.

Abstract

We investigate the numerical solution of multiscale transport equations using Physics Informed Neural Networks (PINNs) with ReLU activation functions. Therefore, we study the analogy between PINNs and Least-Squares Finite Elements (LSFE) which lies in the shared approach to reformulate the PDE solution as a minimization of a quadratic functional. We prove that in the diffusive regime, the correct limit is not reached, in agreement with known results for first-order LSFE. A diffusive scaling is introduced that can be applied to overcome this, again in full agreement with theoretical results for LSFE. We provide numerical results in the case of slab geometry that support our theoretical findings.