On solving/learning differential equations with kernels

Abstract: We present a simple, rigorous, and unified framework for solving and learning (possibly nonlinear) differential equations (PDEs and ODEs) using the framework of Gaussian processes/kernel methods. For PDEs the proposed approach:
(1) provides a natural generalization of collocation kernel methods to nonlinear PDEs and Inverse Problems;
(2) has guaranteed convergence for a very general class of PDEs, and comes equipped with a path to compute error bounds for specific PDE approximations;
(3) inherits the state-of-the-art computational complexity of linear solvers for dense kernel matrices.
For ODEs, we illustrate the efficacy of the proposed approach by extrapolating weather/climate time series obtained from satellite data and illustrate the importance of using adapted/learned kernels.
Parts of this talk are joint work with Yifan Chen, Boumediene Hamzi, Bamdad Hosseini, Romit Maulik, Florian Schäfer, Clint Scovel and Andrew Stuart.

numerical analysis

Audience: researchers in the topic

Seminars on Numerics and Applications