Score-based Transport Modeling for Mean-Field Fokker-Planck Equations

TL;DR

This paper introduces MSBTM, a score-based transport method for solving mean-field Fokker-Planck equations, providing theoretical error bounds and validating its effectiveness through numerical experiments.

Contribution

The paper develops MSBTM, a novel score-based transport approach for mean-field Fokker-Planck equations, with theoretical error analysis and comprehensive numerical validation.

Findings

Validated MSBTM against stochastic differential equation solutions

Established an upper bound on KL divergence derivative

Demonstrated effectiveness on different mean-field equations

Abstract

We use the score-based transport modeling method to solve the mean-field Fokker-Planck equations, which we call MSBTM. We establish an upper bound on the time derivative of the Kullback-Leibler (KL) divergence to MSBTM numerical estimation from the exact solution, thus validates the MSBTM approach. Besides, we provide an error analysis for the algorithm. In numerical experiments, we study two types of mean-field Fokker-Planck equation and their corresponding dynamics of particles in interacting systems. The MSBTM algorithm is numerically validated through qualitative and quantitative comparison between the MSBTM solutions, the results of integrating the associated stochastic differential equation and the analytical solutions if available.