Physics-informed machine learning as a kernel method

Nathan Doumèche, Francis Bach, Gérard Biau, Claire Boyer
Proceedings of Thirty Seventh Conference on Learning Theory, PMLR 247:1399-1450, 2024.

Abstract

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 ˆfn of the regularized risk and show that ˆfn 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v247-doumeche24a, title = {Physics-informed machine learning as a kernel method}, author = {Doum{\`e}che, Nathan and Bach, Francis and Biau, G{\'e}rard and Boyer, Claire}, booktitle = {Proceedings of Thirty Seventh Conference on Learning Theory}, pages = {1399--1450}, year = {2024}, editor = {Agrawal, Shipra and Roth, Aaron}, volume = {247}, series = {Proceedings of Machine Learning Research}, month = {30 Jun--03 Jul}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v247/doumeche24a/doumeche24a.pdf}, url = {https://proceedings.mlr.press/v247/doumeche24a.html}, abstract = {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 $\hat f_n$ of the regularized risk and show that $\hat f_n$ 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.} }
Endnote
%0 Conference Paper %T Physics-informed machine learning as a kernel method %A Nathan Doumèche %A Francis Bach %A Gérard Biau %A Claire Boyer %B Proceedings of Thirty Seventh Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2024 %E Shipra Agrawal %E Aaron Roth %F pmlr-v247-doumeche24a %I PMLR %P 1399--1450 %U https://proceedings.mlr.press/v247/doumeche24a.html %V 247 %X 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 $\hat f_n$ of the regularized risk and show that $\hat f_n$ 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.
APA
Doumèche, N., Bach, F., Biau, G. & Boyer, C.. (2024). Physics-informed machine learning as a kernel method. Proceedings of Thirty Seventh Conference on Learning Theory, in Proceedings of Machine Learning Research 247:1399-1450 Available from https://proceedings.mlr.press/v247/doumeche24a.html.

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