Unified generalization analysis for physics informed neural networks

2026-05-14 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Scientific Computing · Depth: Expert, extended

Summary

This paper presents a unified generalization analysis for Physics-Informed Neural Networks (PINNs) and Variational Physics-Informed Neural Networks (VPINNs), addressing limitations in existing analyses that often require restrictive assumptions. The authors derive novel generalization bounds for neural networks that incorporate differentiation with respect to input variables. By employing Taylor expansion to represent nonlinear differential operators as linear operators in a high-dimensional space, the study extends Koopman-based analysis, demonstrating that high-rank networks can generalize effectively even with differential operators. The research also highlights that the nonlinearity of the differential operator exponentially increases the generalization bound, underscoring its significant impact. Numerical experiments on the 2D Navier-Stokes and Monge-Ampère equations confirm the validity of the proposed bounds and the effectiveness of a regularization strategy derived from them, showing improved test loss for both VPINNs and standard PINNs.

Key takeaway

For AI Scientists and Research Scientists developing or deploying Physics-Informed Neural Networks, understanding the impact of differential operator nonlinearity on generalization is critical. Your models can achieve better test performance by incorporating regularization terms derived from Koopman-based bounds, which favor high-rank weight matrices. Consider VPINNs over standard PINNs for improved generalization properties, especially when dealing with highly nonlinear physical systems.

Key insights

Koopman-based analysis can unify generalization bounds for PINNs and VPINNs, even with nonlinear differential operators.

Principles

• High-rank networks can generalize well with differential operators.
• Nonlinearity of differential operators exponentially enlarges generalization bounds.

Method

Taylor expansion transforms nonlinear differential operators into linear operators on a higher-dimensional space, enabling Koopman-based generalization analysis for PINNs/VPINNs.

In practice

• Regularization terms can be designed to minimize generalization bounds.
• VPINNs offer generalization advantages over standard PINNs.

Topics

• Physics-Informed Neural Networks
• Variational PINNs
• Generalization Bounds
• Koopman Operators
• Nonlinear Differential Operators

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.