Simulating Fokker-Planck equations via mean field control of score-based normalizing flows

2025/06/06 by Mo Zhou, Stanley Osher, Zhou, Mo +3
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #34K35 #37N35 #58E25 #93C15 #93C20 #FOS: Mathematics #G.1.7 #Markov Chains and Monte Carlo Methods #Optimization and Control (math.OC) #Stochastic processes and financial applications #acm:34K35 #acm:37N35 #acm:58E25 #acm:93C15 #acm:93C20 #math.OC #msc:34K35 #msc:37N35 #msc:58E25 #msc:93C15 #msc:93C20 #stochastic dynamics and bifurcation

paper · pdf · doi:10.48550/arxiv.2506.05723

openalex publication_date 2025/06/06 · openalex created_date 2025/10/10 · arxiv created 2026/07/29 · arxiv updated 2026/07/31 · openalex updated_date 2026/08/03

Abstract

The Fokker--Planck (FP) equation governs the evolution of densities for stochastic dynamics of physical systems, such as the Langevin dynamics and the Lorenz system. This work simulates FP equations through a mean field control (MFC) problem. We first formulate the FP equation as a continuity equation, where the velocity field consists of the drift function and the score function, i.e., the gradient of the logarithm of the density function. Next, we design a MFC problem that matches the velocity fields in a continuity equation with the ones in the FP equation. The score functions along deterministic trajectories are computed efficiently through the score-based normalizing flow, which only relies on the derivatives of the parameterized velocity fields. Numerical results, including Langevin dynamics, underdamped Langevin dynamics, chaotic systems, and high-dimensional interacting particle systems validate the effectiveness and scalability of our proposed algorithm. A convergence analysis is conducted for our algorithm on the FP equation of Ornstein--Uhlenbeck processes.

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Related

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