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

arXiv: https://arxiv.org/abs/2510.23149 · HTML: https://mlsanity.com/paper/2510.23149/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 46 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 15 of 15)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Informed Machine Learning -- A Taxonomy and Survey of Integrating Knowledge into Learning Systems](https://mlsanity.com/paper/1903.12394/) (2019)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [When Physics Meets Machine Learning: A Survey of Physics-Informed Machine Learning](https://mlsanity.com/paper/2203.16797/) (2022)
- [Error estimates for physics informed neural networks approximating the Navier-Stokes equations](https://mlsanity.com/paper/2203.09346/) (2022)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Error analysis for physics informed neural networks (PINNs) approximating Kolmogorov PDEs](https://mlsanity.com/paper/2106.14473/) (2021)
- [Learning without Concentration](https://mlsanity.com/paper/1401.0304/) (2014)
- [Relative Deviation Learning Bounds and Generalization with Unbounded Loss Functions](https://mlsanity.com/paper/1310.5796/) (2013)
- [Learning without Concentration for General Loss Functions](https://mlsanity.com/paper/1410.3192/) (2014)
- [Physics-informed machine learning as a kernel method](https://mlsanity.com/paper/2402.07514/) (2024)
- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Physics-informed learning under mixing: How physical knowledge speeds up learning](https://mlsanity.com/paper/2509.24801/) (2025)
- [Understanding Generalization in Physics Informed Models through Affine Variety Dimensions](https://mlsanity.com/paper/2501.18879/) (2025)

## Built on by (most cited first, 1 of 1)

- [Random test functions, H-1 norm equivalence, and stochastic variational physics-informed neural networks](https://mlsanity.com/paper/2605.03542/) (2026)

## 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: (A) what is the trade-off between soft penalties and hard constraints, and (B) 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 classes of functions whose convex hull satisfies a small-ball condition, in particular convex classes, under physical knowledge expressed as linear equations by developing appropriate complexity dependent error rates based on the small-ball method. We specialise them to linear classes of finite dimension q in which the space of solutions of the equation has dimension k < q, and show that (A) the physics-informed estimator attains the error bound of the hard-constrained empirical error minimiser whenever the penalty parameter is above an explicit threshold, and that (B) the penalty improves the error rate from √q/n to √k/n under a second moment condition on the noise only, having the effect of a dimensionality reduction. We also quantify how these conclusions degrade when the equation is misspecified, in terms of the excess residual of the target function. This work establishes a theoretical framework for evaluating the statistical properties of physics-informed estimators in such classes, contributing to closing the gap between statistical theory and practical PISL.