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

2025-03-14

AI Summary

Efficiently solving the high-dimensional Fokker–Planck equation (FPE) for complex stochastic dynamical systems remains challenging due to the curse of dimensionality. Method: We propose a mesh-free, adaptive neural PDE solver that reformulates the FPE via the Feynman–Kac formula as a path-integral expectation estimation. Our approach innovatively integrates temporal normalizing flows with an adaptive importance sampling mechanism to explicitly model time-varying probability densities. Contribution/Results: The framework circumvents the curse of dimensionality, achieving both accuracy and efficiency in arbitrarily high dimensions. Experiments demonstrate substantial improvements over state-of-the-art methods across diverse high-dimensional stochastic differential equation (SDE) benchmarks; notably, it exhibits strong robustness, scalability, and computational efficiency in settings exceeding three dimensions. This work establishes a novel paradigm for modeling high-dimensional non-stationary stochastic systems.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Intelligent Robots: State Estimation
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

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.

Problem

• Solving high-dimensional Fokker-Planck equations efficiently
• Reducing computational complexity with adaptive stochastic sampling
• Enhancing accuracy in complex dynamical systems

Innovation

• Uses Feynman-Kac formula for solution queries
• Implements adaptive stochastic sampling scheme
• Employs time-indexed normalizing flow for densities