Conference Agenda

Physics-informed machine learning typically integrates physical priors into the learning process by minimizing a loss function that combines a data-driven term with partial differential equation regularization. Practitioners often rely on physics-informed neural networks (PINNs) to address this type of problem. After discussing the strengths and limitations of PINNs, I will show that for linear differential priors, the problem can be directly formulated as a kernel regression task, providing a rigorous framework for analyzing physics-informed machine learning. In particular, incorporating the physical prior can significantly enhance the estimator's convergence.

Building on this kernel regression formulation, I will explain how Fourier methods can be used to approximate the associated kernel and propose a tractable estimator that minimizes the physics-informed risk function. This approach, which we refer to as physics-informed kernel learning (PIKL), provides theoretical guarantees on performance. We will demonstrate the numerical performance of the PIKL estimator through simulations in both hybrid modeling and PDE-solving contexts.

Joint work with Francis Bach (Inria), Claire Boyer (University Paris-Saclay), and Nathan Doumèche (Sorbonne University).