  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.
