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Fokker-Planck equations are extensively employed in various scientific fieldsas they characterise the behaviour of stochastic systems at the level ofprobability density functions. Although broadly used, they allow for analyticaltreatment only in limited settings, and often is inevitable to resort tonumerical solutions. Here, we develop a computational approach for simulatingthe time evolution of Fokker-Planck solutions in terms of a mean field limit ofan interacting particle system. The interactions between particles aredetermined by the gradient of the logarithm of the particle density,approximated here by a novel statistical estimator. The performance of ourmethod shows promising results, with more accurate and less fluctuatingstatistics compared to direct stochastic simulations of comparable particlenumber. Taken together, our framework allows for effortless and reliableparticle-based simulations of Fokker-Planck equations in low and moderatedimensions. The proposed gradient-log-density estimator is also of independentinterest, for example, in the context of optimal control.

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