Self-Consistency of the Fokker-Planck Equation

TL;DR

This paper introduces a self-consistency framework for the Fokker-Planck equation, enabling neural network-based modeling of density evolution and proving convergence of the generated trajectories to the true solution.

Contribution

It proposes a potential function for hypothesis velocity fields and demonstrates convergence of neural network models to the FPE solution using this framework.

Findings

Potential function diminishes during training leading to convergence.

Neural network parameterization allows efficient computation of gradients.

Generated trajectories approximate the true FPE solution in Wasserstein-2 sense.

Abstract

The Fokker-Planck equation (FPE) is the partial differential equation that governs the density evolution of the It\^o process and is of great importance to the literature of statistical physics and machine learning. The FPE can be regarded as a continuity equation where the change of the density is completely determined by a time varying velocity field. Importantly, this velocity field also depends on the current density function. As a result, the ground-truth velocity field can be shown to be the solution of a fixed-point equation, a property that we call self-consistency. In this paper, we exploit this concept to design a potential function of the hypothesis velocity fields, and prove that, if such a function diminishes to zero during the training procedure, the trajectory of the densities generated by the hypothesis velocity fields converges to the solution of the FPE in the Wasserstein-2 sense. The proposed potential function is amenable to neural-network based parameterization as the stochastic gradient with respect to the parameter can be efficiently computed. Once a parameterized model, such as Neural Ordinary Differential Equation is trained, we can generate the entire trajectory to the FPE.