# FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing flows and the Feynman Kac-Formula

**Authors:** Naoufal El Bekri, Lucas Drumetz, Franck Vermet  
**arXiv:** [2503.11427](https://arxiv.org/abs/2503.11427)  
**Date:** 2025-03-14  
**Summary page:** https://www.summarizepaper.com/en/paper/2503.11427v1/

## Key points

- Solving the Fokker-Planck equation in high-dimensional complex dynamical systems is challenging due to lack of analytical solutions and limitations of traditional numerical methods.
- FlowKac is a groundbreaking approach that reformulates the Fokker-Planck equation using the Feynman-Kac formula, allowing queries at specific points through expected values of stochastic paths.
- FlowKac introduces an adaptive stochastic sampling scheme to reduce computational complexity while maintaining accuracy, along with a time-indexed normalizing flow for capturing time-evolving probability densities.
- The method facilitates robust sampling of collocation points, offering a flexible and mesh-free solver that mitigates the curse of dimensionality.
- FlowKac enhances computational accuracy and efficiency for applications beyond conventional three dimensions, demonstrating significant improvements over existing techniques in experiments on stochastic differential equations.

## Summary

In the realm of high-dimensional complex dynamical systems, solving the Fokker-Planck equation has long been a challenging yet essential task. This is due to the lack of analytical solutions and limitations of traditional numerical methods. However, a groundbreaking approach known as FlowKac has been introduced to revolutionize the solution process. It reformulates the Fokker-Planck equation using the Feynman-Kac formula and enables queries of the solution at specific points through expected values of stochastic paths. One of its key innovations is an adaptive stochastic sampling scheme that effectively reduces computational complexity while maintaining high accuracy. This is complemented by a time-indexed normalizing flow designed to capture time-evolving probability densities. Together, these components facilitate robust sampling of collocation points, resulting in a flexible and mesh-free solver that mitigates the curse of dimensionality. The significance of FlowKac extends beyond efficiency; it enhances computational accuracy and efficiency for applications requiring dimensions beyond conventional three. Through various experiments on stochastic differential equations, FlowKac has demonstrated remarkable robustness and scalability, showcasing significant improvements over existing techniques. By providing a more detailed understanding of complex dynamical systems and offering precise solutions in high-dimensional spaces, FlowKac stands as a pioneering method in advancing computational approaches for intricate problems in diverse fields.