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Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method

arXiv 2510.23149 ↗First posted 27 Oct 2025stat.ML cs.LG math.STcited by 0

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Abstract (arXiv)

Physics-informed statistical learning (PISL) integrates empirical data with physical knowledge to enhance the statistical performance of estimators. While PISL methods are widely used in practice, a comprehensive theoretical understanding of how informed regularization affects statistical properties is still missing. Specifically, two fundamental questions have yet to be fully addressed: (1) what is the trade-off between considering soft penalties versus hard constraints, and (2) what is the statistical gain of incorporating physical knowledge compared to purely data-driven empirical error minimisation. In this paper, we address these questions for PISL in convex classes of functions under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We show that, under suitable assumptions, (1) the error rates of physics-informed estimators are comparable to those of hard constrained empirical error minimisers, differing only by constant terms, and that (2) informed penalization can effectively reduce model complexity, akin to dimensionality reduction, thereby improving learning performance. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in convex classes of functions, contributing to closing the gap between statistical theory and practical PISL, with potential applications to cases not yet explored in the literature.

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References · 15

  1. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  2. 2019Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systemscited by 203
  3. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  4. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  5. 2022When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learningcited by 64
  6. 2022Error estimates for physics informed neural networks approximating the Navier-Stokes equationscited by 63
  7. 2021When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?cited by 57
  8. 2021Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEscited by 56
  9. 2014Learning without Concentrationcited by 50
  10. 2013Relative Deviation Learning Bounds and Generalization with Unbounded Loss Functionscited by 15
  11. 2014Learning without Concentration for General Loss Functionscited by 13
  12. 2024Physics-informed machine learning as a kernel methodcited by 11
3 more
  1. 2024Physics-informed kernel learningcited by 11
  2. 2025Physics-informed learning under mixing: How physical knowledge speeds up learningcited by 2
  3. 2025Understanding Generalization in Physics Informed Models through Affine Variety Dimensionscited by 1

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