Dynamics for sampling: kinetic equations, gradient flows and beyond

Sampling from a target probability distribution is crucial across scientific domains, such as molecular dynamics, statistical physics, Bayesian statistics, and machine learning. The target distribution encodes critical information about systems, such as the likelihood of particle configurations in physical space or parameter choices in large dataset models. Sampling algorithms simulate particle evolution over time, converging to the target distribution after a long time. The PI aims to advance his understanding of the efficiency of these sampling algorithms and enhance their performance, providing useful insights for practitioners. The PI will mentor undergraduate researchers and develop courses on sampling for advanced undergraduate and early-stage graduate students. Mathematical tools from partial and stochastic differential equations, probability theory, and numerical analysis will be used to study both theoretical and algorithmic aspects of sampling dynamics. Specifically, the project will: (1) develop a framework using functional inequalities, Sobolev spaces, and Markov semigroup theory to analyze long-time convergence properties of sampling dynamics characterized by kinetic equations, and (2) use techniques from optimization, optimal transport, applied analysis, and mean-field limits of interacting particle systems to study sampling dynamics structured as gradient flows, including scenarios where standard gradient flow theory is not directly applicable. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.