Claire Boyer est professeur au LMO, Université Paris-Saclay.
Titre : A primer on physics-informed machine learning
Résumé : 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. Practitioners often resort to physics-informed neural networks (PINNs) to solve this kind of problem. After discussing some strengths and limitations of PINNs, we prove that for linear differential priors, the problem can be formulated directly as a kernel regression task, giving a rigorous framework to analyze physics-informed ML. In particular, the physical prior can help in boosting the estimator convergence. We also propose the PIKL algorithm (PIKL for physics-informed kernel learning) as a numerical strategy to implement this kernel method.