Weak Generative Sampler for Stationary Distributions of McKean-Vlasov System

Abstract: Stochastic interacting particle systems are widely used to model collective phenomena across diverse fields, including statistical physics, biology, and social dynamics. The McKean-Vlasov equation arises as the mean-field limit of such systems as the number of particles tends to infinity, while its long-time behaviour is characterized by stationary distributions as time tends to infinity. However, the validity of interchanging the infinite-time and infinite-particle limits is not guaranteed. Consequently, simulation methods that rely on a finite-particle truncation may fail to accurately capture the mean-field system's stationary distributions, particularly when the coexistence of multiple metastable states leads to phase transitions. In this paper, we adapt the framework of the Weak Generative Sampler (WGS) -- a generative technique based on normalizing flows and a weak formulation of the nonlinear Fokker-Planck equation -- to compute and generate i.i.d. samples satisfying the stationary distributions of McKean-Vlasov processes. Extensive numerical experiments validate the efficacy of the proposed methods, showcasing their ability to accurately approximate stationary distributions and capture phase transitions in complex systems.