Unified generalization analysis for physics informed neural networks

TL;DR

This paper develops a unified theoretical framework to analyze the generalization capabilities of physics-informed neural networks, accounting for nonlinear differential operators and high-dimensional input spaces.

Contribution

It introduces a novel approach using Taylor expansion and Koopman analysis to derive generalization bounds for PINNs and VPINNs under minimal assumptions.

Findings

High-rank neural networks can generalize well with differential operators.

Nonlinearity of differential operators exponentially increases the generalization bound.

Unified analysis applies to both PINNs and VPINNs without restrictive assumptions.

Abstract

Physics-Informed Neural Networks (PINNs) and their variational counterparts (VPINNs) are neural networks that incorporate physical laws, making them useful for scientific problems. Existing generalization analyses for PINNs and VPINNs remain limited, often requiring restrictive assumptions such as stability conditions or linear ellipticity. In this paper, we derive generalization bounds for neural networks that involve differentiation with respect to input variables, covering PINNs and VPINNs under a unified framework. We apply Taylor expansion to represent nonlinear differential operators as linear operators on a high-dimensional space, enabling the use of Koopman-based analysis and showing that high-rank networks can generalize well even in settings involving differential operators. We also show that the nonlinearity of the differential operator exponentially enlarges the bound, highlighting its significant impact on generalization.