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

arXiv: https://arxiv.org/abs/2006.00702 · HTML: https://mlsanity.com/paper/2006.00702/

## Key facts

- Cited by 88 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2020.

## Built on by (most cited first, 30 of 88)

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- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
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- [Solving Inverse Physics Problems with Score Matching](https://mlsanity.com/paper/2301.10250/) (2023)
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- [Wasserstein proximal operators describe score-based generative models and resolve memorization](https://mlsanity.com/paper/2402.06162/) (2024)
- [Stein transport for Bayesian inference](https://mlsanity.com/paper/2409.01464/) (2024)
- [Physics-Informed Distillation of Diffusion Models for PDE-Constrained Generation](https://mlsanity.com/paper/2505.22391/) (2025)
- [Deterministic Langevin Monte Carlo with Normalizing Flows for Bayesian Inference](https://mlsanity.com/paper/2205.14240/) (2022)
- [What makes an image realistic?](https://mlsanity.com/paper/2403.04493/) (2024)
- [InvFusion: Bridging Supervised and Zero-shot Diffusion for Inverse Problems](https://mlsanity.com/paper/2504.01689/) (2025)

## Abstract (arXiv)

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.