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.
| Subjects: | Machine Learning (cs.LG); Analysis of PDEs (math.AP); Functional Analysis (math.FA); Machine Learning (stat.ML) |
| Cite as: | arXiv:2605.13260 [cs.LG] |
| (or arXiv:2605.13260v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.13260 arXiv-issued DOI via DataCite (pending registration) |
Submission history
From: Yuka Hashimoto [view email]
[v1]
Wed, 13 May 2026 09:42:33 UTC (302 KB)