Abstract: We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. We establish connections between MFGs and major classes of flow and diffusion-based generative models by deriving continuous-time normalizing flows, score-based models, and Wasserstein gradient flows through different choices of particle dynamics and cost functions. Furthermore, we study the mathematical structure and properties of each generative model by examining their associated MFG's optimality condition, which consist of a set of coupled forward-backward nonlinear partial differential equations. The optimality conditions of MFGs also allow us to introduce HJB regularizers for enhanced training of a broad class of generative models. We present this framework as an MFG laboratory which serves as a platform for revealing new avenues of experimentation and invention of generative models.
Bio: Benjamin Zhang is a postdoctoral research associate in the Department of Mathematics and Statistics at the University of Massachusetts Amherst working with Markos Katsoulakis, Luc-Rey Bellet, and Paul Dupuis. His research interests lie at the intersection of computational statistics and computational dynamics, with a recent focus on generative modeling and mean-field games. His other interests include sampling methods for rare event simulation and Bayesian inference. He received his PhD in Computational Science and Engineering from MIT in 2022 under the supervision of Youssef Marzouk. Before MIT, he received his Bachelor’s degrees in engineering physics and applied mathematics at UC Berkeley.