This work is concerned with solving high-dimensional Fokker-Planck equations with the novel perspective that solving the PDE can be reduced to independent instances of density estimation tasks based on the trajectories sampled from its associated particle dynamics. With this approach, one sidesteps error accumulation occurring from integrating the PDE dynamics on a parameterized function class. This approach significantly simplifies deployment, as one is free of the challenges of implementing loss terms based on the differential equation. In particular, we introduce a novel class of high-dimensional functions called the functional hierarchical tensor (FHT). The FHT ansatz leverages a hierarchical low-rank structure, offering the advantage of linearly scalable runtime and memory complexity relative to the dimension count. We introduce a sketching-based technique that performs density estimation over particles simulated from the particle dynamics associated with the equation, thereby obtaining a representation of the Fokker-Planck solution in terms of our ansatz. We apply the proposed approach successfully to three challenging time-dependent Ginzburg-Landau models with hundreds of variables.

翻译：本文致力于求解高维Fokker-Planck方程，提出了一种新颖视角：求解该偏微分方程可简化为基于其相关粒子动力学采样轨迹的密度估计任务的独立实例。该方法避免了在参数化函数类上积分偏微分方程动力学时出现的误差累积。由于无需处理基于微分方程的损失项实现难题，该方法显著简化了部署流程。具体而言，我们引入了一类称为函数化层级张量（FHT）的新型高维函数。FHT拟设利用层级低秩结构，具有随维度数量线性扩展的运行时间和内存复杂度的优势。我们提出基于素描的技术，对从方程相关粒子动力学模拟的粒子进行密度估计，从而通过拟设获得Fokker-Planck解的表示。我们成功将所提方法应用于三个包含数百个变量的挑战性含时Ginzburg-Landau模型。