  This work proposes a novel numerical scheme for solving the high-dimensional
Hamilton-Jacobi-Bellman equation with a functional hierarchical tensor ansatz.
We consider the setting of stochastic control, whereby one applies control to a
particle under Brownian motion. In particular, the existence of diffusion
presents a new challenge to conventional tensor network methods for
deterministic optimal control. To overcome the difficulty, we use a general
regression-based formulation where the loss term is the Bellman consistency
error combined with a Sobolev-type penalization term. We propose two novel
sketching-based subroutines for obtaining the tensor-network approximation to
the action-value functions and the value functions, which greatly accelerate
the convergence for the subsequent regression phase. We apply the proposed
approach successfully to two challenging control problems with Ginzburg-Landau
potential in 1D and 2D with 64 variables.
