Interacting particle solutions of Fokker-Planck equations through gradient-log-density estimation

TL;DR

This paper introduces a particle-based computational method for solving Fokker-Planck equations using a novel gradient-log-density estimator, enabling more accurate and stable simulations in low to moderate dimensions.

Contribution

It presents a new approach combining interacting particles and a statistical estimator for gradient-log-density, improving simulation accuracy over traditional stochastic methods.

Findings

More accurate statistical estimates compared to direct stochastic simulations.

Reduced fluctuations in particle-based simulations.

Effective for low and moderate-dimensional Fokker-Planck problems.

Abstract

Fokker-Planck equations are extensively employed in various scientific fields as they characterise the behaviour of stochastic systems at the level of probability density functions. Although broadly used, they allow for analytical treatment only in limited settings, and often is inevitable to resort to numerical solutions. Here, we develop a computational approach for simulating the time evolution of Fokker-Planck solutions in terms of a mean field limit of an interacting particle system. The interactions between particles are determined by the gradient of the logarithm of the particle density, approximated here by a novel statistical estimator. The performance of our method shows promising results, with more accurate and less fluctuating statistics compared to direct stochastic simulations of comparable particle number. Taken together, our framework allows for effortless and reliable particle-based simulations of Fokker-Planck equations in low and moderate dimensions. The proposed gradient-log-density estimator is also of independent interest, for example, in the context of optimal control.