FlowKac: An Efficient Neural Fokker-Planck solver using Temporal Normalizing Flows and the Feynman-Kac Formula
This addresses a challenging problem in computational physics and machine learning for high-dimensional dynamical systems, offering an incremental improvement over existing methods.
The paper tackles solving high-dimensional Fokker-Planck equations by introducing FlowKac, which uses the Feynman-Kac formula and adaptive stochastic sampling with time-indexed normalizing flows, resulting in reduced computational complexity and improved accuracy for stochastic differential equations.
Solving the Fokker-Planck equation for high-dimensional complex dynamical systems remains a pivotal yet challenging task due to the intractability of analytical solutions and the limitations of traditional numerical methods. In this work, we present FlowKac, a novel approach that reformulates the Fokker-Planck equation using the Feynman-Kac formula, allowing to query the solution at a given point via the expected values of stochastic paths. A key innovation of FlowKac lies in its adaptive stochastic sampling scheme which significantly reduces the computational complexity while maintaining high accuracy. This sampling technique, coupled with a time-indexed normalizing flow, designed for capturing time-evolving probability densities, enables robust sampling of collocation points, resulting in a flexible and mesh-free solver. This formulation mitigates the curse of dimensionality and enhances computational efficiency and accuracy, which is particularly crucial for applications that inherently require dimensions beyond the conventional three. We validate the robustness and scalability of our method through various experiments on a range of stochastic differential equations, demonstrating significant improvements over existing techniques.