Is $L^2$ Physics-Informed Loss Always Suitable for Training Physics-Informed Neural Network?

TL;DR

This paper challenges the standard use of $L^2$ loss in training PINNs for certain PDEs, showing that $L^{}$ loss is more suitable for stability and proposing a new training algorithm accordingly.

Contribution

It provides a theoretical analysis linking loss function choice to stability in PINNs and introduces an $L^{}$ loss-based training method for HJB equations.

Findings

$L^{}$ loss improves stability for HJB equations.

Standard $L^2$ loss may be unsuitable for certain high-dimensional PDEs.

Proposed $L^{}$ loss training algorithm outperforms traditional methods.

Abstract

The Physics-Informed Neural Network (PINN) approach is a new and promising way to solve partial differential equations using deep learning. The $L^2$ Physics-Informed Loss is the de-facto standard in training Physics-Informed Neural Networks. In this paper, we challenge this common practice by investigating the relationship between the loss function and the approximation quality of the learned solution. In particular, we leverage the concept of stability in the literature of partial differential equation to study the asymptotic behavior of the learned solution as the loss approaches zero. With this concept, we study an important class of high-dimensional non-linear PDEs in optimal control, the Hamilton-Jacobi-Bellman(HJB) Equation, and prove that for general $L^p$ Physics-Informed Loss, a wide class of HJB equation is stable only if $p$ is sufficiently large. Therefore, the commonly used $L^2$ loss is not suitable for training PINN on those equations, while $L^{\infty}$ loss is a better choice. Based on the theoretical insight, we develop a novel PINN training algorithm to minimize the $L^{\infty}$ loss for HJB equations which is in a similar spirit to adversarial training. The effectiveness of the proposed algorithm is empirically demonstrated through experiments. Our code is released at https://github.com/LithiumDA/L_inf-PINN.