Generating Samples of Stationary Distributions of Weakly Interacting Diffusion Models Without Finite Particle Truncation: A Weak Generative Approach

Abstract

Computing the stationary probability density and generating corresponding samples for the mean-field model of an infinite number of weakly interacting diffusion particles pose significant numerical challenges, particularly in the phase transition regime where the interchangeability of infinite-time and infinite-particle limits breaks down. Traditional approaches, such as direct simulation of finite-particle systems, often fail to accurately pinpoint multiple stationary distributions in the mean-field meta-stable setting. On the other hand, solving the high-dimensional McKean-Vlasov partial differential equation using neural networks typically yields only the density function, limiting its utility for estimating statistical quantities from generating samples. In this work, we propose a novel generative framework based on the weak PDE formulation of the mean-field model to address these challenges. Our approach simultaneously computes the stationary distributions of McKean-Vlasov processes and generates independent and identically distributed samples that satisfy these distributions. This integrated approach not only reveals the true stationary distributions without the random perturbation of finite particle truncation, but also offers deeper insight into the system's behavior in the mean-field limit. Extensive numerical experiments demonstrate the effectiveness of the proposed method, showcasing its ability to accurately approximate stationary distributions, capture intricate phase transitions, and handle high-dimensional complex systems.