Understanding Generalization in Physics Informed Models through Affine Variety Dimensions
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Abstract (arXiv)
Physics-informed machine learning is gaining significant traction for enhancing statistical performance and sample efficiency through the integration of physical knowledge. However, current theoretical analyses often presume complete prior knowledge in non-hybrid settings, overlooking the crucial integration of observational data, and are frequently limited to linear systems, unlike the prevalent nonlinear nature of many real-world applications. To address these limitations, we introduce a unified residual form that unifies collocation and variational methods, enabling the incorporation of incomplete and complex physical constraints in hybrid learning settings. Within this formulation, we establish that the generalization performance of physics-informed regression in such hybrid settings is governed by the dimension of the affine variety associated with the physical constraint, rather than by the number of parameters. This enables a unified analysis that is applicable to both linear and nonlinear equations. We also present a method to approximate this dimension and provide experimental validation of our theoretical findings.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 16 of its 42 references that are in the map. First posted on arXiv 31 Jan 2025.
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References · 16
- 2019Optuna: A Next-generation Hyperparameter Optimization Frameworkcited by 2,455
- 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
- 2019Towards Physics-informed Deep Learning for Turbulent Flow Predictioncited by 203
- 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
- 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
- 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
- 2022Physics-Informed Machine Learning: A Survey on Problems, Methods and Applicationscited by 88
- 2019A Theoretical Analysis of Deep Neural Networks and Parametric PDEscited by 74
- 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
- 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
- 2020Physics Informed Neural Networks for Simulating Radiative Transfercited by 35
- 2023Lie Point Symmetry and Physics Informed Networkscited by 24
4 more
- 2022AutoIP: A United Framework to Integrate Physics into Gaussian Processescited by 20
- 2024Physics-informed machine learning as a kernel methodcited by 11
- 2024Physics-informed kernel learningcited by 11
- 2023Covering Number of Real Algebraic Varieties and Beyond: Improved Bounds and Applicationscited by 7
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