Applied Mathematics Colloquia by Fei Lu: Statistical Learning and Inverse Problems for Interacting Particle Systems

Speaker: Fei Lu, associate professor of mathematics, Johns Hopkins University

Title: Statistical Learning and Inverse Problems for Interacting Particle Systems

Abstract:

Systems of 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 learn the rules of interaction from data. We study the nonparametric regression estimator for the pairwise interaction kernels from trajectory data of differential systems, with examples including opinion dynamics, the Lennard-Jones system, and mean-field PDE. When the system has finite particles, we have a statistical learning problem, and we provide a systematic learning theory addressing the fundamental issues, such as identifiability and mini-max convergence rate for the estimator. When the system has infinite particles, we have an ill-posed inverse problem for PDEs, and we introduce a new regularization method using adaptive RKHSs. Furthermore, learning kernels in operators emerges as a new topic unifying the two settings, and we discuss related open questions.

Applied Mathematics Colloquium