# Francis Bach

## Researcher at Inria

# Physics-informed Machine Learning as a Kernel Method

## 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 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. (joint work with Nathan Doumèche, Gérard Biau, and Claire Boyer).

### About Francis Bach

Francis Bach is a researcher at Inria, leading since 2011 the machine learning team which is part of the Computer Science department at Ecole Normale Supérieur. He graduated from Ecole Polytechnique in 1997 and completed his Ph.D. in Computer Science at U.C. Berkeley in 2005, working with Professor Michael Jordan. He spent two years in the Mathematical Morphology group at Ecole des Mines de Paris, then he joined the computer vision project-team at Inria/Ecole Normale Supérieur from 2007 to 2010.

Francis Bach is primarily interested in machine learning, and especially in sparse methods, kernel-based learning, neural networks, and large-scale optimization. He obtained in 2009 a Starting Grant and in 2016 a Consolidator Grant from the European Research Council, and received the INRIA young researcher prize in 2012, the ICML test-of-time award in 2014 and 2019, the NeurIPS test-of-time award in 2021, as well as the Lagrange prize in continuous optimization in 2018, and the Jean-Jacques Moreau prize in 2019. He was elected in 2020 at the French Academy of Sciences. In 2015, he was program co-chair of the International Conference in Machine learning (ICML), general chair in 2018, and president of its board between 2021 and 2023; he was co-editor-in-chief of the Journal of Machine Learning Research between 2018 and 2023.