Unified generalization analysis for physics informed neural networks

2026-05-13

AI Summary

This work addresses the lack of a unified and relaxed theoretical framework for generalization analysis in physics-informed neural networks (PINNs) and their variational counterparts (VPINNs), which has previously relied on strong assumptions. The authors propose a high-dimensional linearization approach based on Taylor expansion to transform nonlinear differential operators into linear operators in an augmented space, integrating Koopman operator theory to establish, for the first time, a unified generalization analysis framework applicable to both PINNs and VPINNs. This framework reveals an exponential amplification effect of nonlinearity on generalization error and demonstrates that high-rank networks exhibit superior generalization capabilities in tasks involving differential operators, offering new theoretical insights for the understanding and design of physics-informed neural networks.

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.

Problem

• Physics-Informed Neural Networks
• generalization analysis
• differential operators
• nonlinearity
• VPINNs

Innovation

• Physics-Informed Neural Networks
• Generalization Bounds
• Koopman Operator
• Taylor Expansion
• Differential Operators