Title: Statistical learning and inverse problems for interacting particle systems
Abstract: 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 learn 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. We show that this inverse problem has a data-dependent function space of identifiability (FSOI). Thus, when the inverse problem is ill-posed, regularization has a new challenge of ensuring that the learning takes place inside the FSOI. We address this challenge by introducing a data-adaptive RKHS Tikhonov regularization (DARTR) method.
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