# Physics-informed machine learning as a kernel method

arXiv: https://arxiv.org/abs/2402.07514 · HTML: https://mlsanity.com/paper/2402.07514/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Feb 2024.

## Built on by (most cited first, 13 of 13)

- [Physics-informed kernel learning](https://mlsanity.com/paper/2409.13786/) (2024)
- [Position: Biology is the Challenge Physics-Informed ML Needs to Evolve](https://mlsanity.com/paper/2510.25368/) (2025)
- [Fast kernel methods: Sobolev, physics-informed, and additive models](https://mlsanity.com/paper/2509.02649/) (2025)
- [Uncertainty-Aware Diagnostics for Physics-Informed Machine Learning](https://mlsanity.com/paper/2510.26121/) (2025)
- [Forecasting time series with constraints](https://mlsanity.com/paper/2502.10485/) (2025)
- [Understanding Generalization in Physics Informed Models through Affine Variety Dimensions](https://mlsanity.com/paper/2501.18879/) (2025)
- [Complexity Dependent Error Rates for Physics-informed Statistical Learning via the Small-ball Method](https://mlsanity.com/paper/2510.23149/) (2025)
- [Benign overfitting in Fixed Dimension via Physics-Informed Learning with Smooth Inductive Bias](https://mlsanity.com/paper/2406.09194/) (2024)
- [Error analysis for the deep Kolmogorov method](https://mlsanity.com/paper/2508.17167/) (2025)
- [A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning](https://mlsanity.com/paper/2605.26341/) (2026)
- [Fast Learning Rates for Physics-Informed Kernel Methods](https://mlsanity.com/paper/2609.18901/) (2026)
- [When do explicit penalties improve flow matching?](https://mlsanity.com/paper/2610.15689/) (2026)
- [AeroChem: Closed-loop Physics-Informed State Space Modeling for Long-term Chemically-Reactive Air Quality Forecasting--Extended Version](https://mlsanity.com/paper/2610.16630/) (2026)

## Abstract (arXiv)

Physics-informed machine learning combines the expressiveness of data-based approaches with the interpretability of physical models. In this context, we consider a general regression problem where the empirical risk is regularized by a partial differential equation that quantifies the physical inconsistency. We prove that for linear differential priors, the problem can be formulated as a kernel regression task. Taking advantage of kernel theory, we derive convergence rates for the minimizer of the regularized risk and show that it converges at least at the Sobolev minimax rate. However, faster rates can be achieved, depending on the physical error. This principle is illustrated with a one-dimensional example, supporting the claim that regularizing the empirical risk with physical information can be beneficial to the statistical performance of estimators.