An operator preconditioning perspective on training in physics-informed machine learning

TL;DR

This paper analyzes how the conditioning of differential operators affects the training of physics-informed neural networks and demonstrates that preconditioning these operators can significantly improve training efficiency.

Contribution

It introduces an operator preconditioning perspective for understanding and improving the training of physics-informed machine learning models, supported by mathematical analysis and empirical results.

Findings

Ill-conditioned operators hinder training efficiency.

Preconditioning improves the conditioning and accelerates training.

Mathematical analysis explains the impact of operator conditioning.

Abstract

In this paper, we investigate the behavior of gradient descent algorithms in physics-informed machine learning methods like PINNs, which minimize residuals connected to partial differential equations (PDEs). Our key result is that the difficulty in training these models is closely related to the conditioning of a specific differential operator. This operator, in turn, is associated to the Hermitian square of the differential operator of the underlying PDE. If this operator is ill-conditioned, it results in slow or infeasible training. Therefore, preconditioning this operator is crucial. We employ both rigorous mathematical analysis and empirical evaluations to investigate various strategies, explaining how they better condition this critical operator, and consequently improve training.