Probability flow solution of the Fokker-Planck equation

Nicholas Matthew Boffi, Eric Vanden-Eijnden

Keywords: score-based diffusion, high-dimensional scientific computing

Abstract: The method of choice for integrating the time-dependent Fokker-Planck equation in high-dimension is to generate samples from the solution via integration of the associated stochastic differential equation. Here, we introduce an alternative scheme based on integrating an ordinary differential equation that describes the flow of probability. Acting as a transport map, this equation deterministically pushes samples from the initial density onto samples from the solution at any later time. Unlike integration of the stochastic dynamics, the method has the advantage of giving direct access to quantities that are challenging to estimate from trajectories alone, such as the probability current, the density itself, and its entropy. The probability flow equation depends on the gradient of the logarithm of the solution (its "score"), and so is a-priori unknown. To resolve this dependence, we model the score with a deep neural network that is learned on-the-fly by propagating a set of samples according to the instantaneous probability current. We consider several high-dimensional examples from the physics of interacting particle systems to highlight the efficiency and precision of the approach; we find that the method accurately matches analytical solutions computed by hand and moments computed via Monte-Carlo.

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TL;DR: We develop an efficient method to solve the Fokker-Planck equation in high-dimension by learning the score of the solution.