Systems of self-interacting particles/agents arise in multiple disciplines, such as particle systems in physics, flocking birds and migrating cells in biology, and opinion dynamics in social science. An essential task in these applications is to infer the rules of interaction from data. We propose nonparametric regression algorithms to learn the pairwise interaction kernels from trajectory data of differential systems, including ODEs/SDEs and mean-field PDEs. Importantly, we provide a systematic learning theory addressing the fundamental issues, such as identifiability and convergence of the estimators. The algorithms and theory are demonstrated in examples including opinion dynamics, the Lennard-Jones system, and aggregation diffusions. Furthermore, learning kernels in operators emerges as a new topic at the intersection of statistical learning and inverse problems. We introduce a data-adaptive RKHS Tikhonov regularization (DARTR) method to address the ill-posedness and discuss open questions on this new topic.

报告人简介：Fei Lu is an Associate Professor in the Department of Mathematics at Johns Hopkins University. His research interests mainly lie in applied probability and computational mathematics, statistics learning, data assimilation, and inverse problems.