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.

翻译：本文提出了一种新颖的数值方案，用于求解高维 Hamilton-Jacobi-Bellman 方程，其核心是采用函数层次张量拟设。我们考虑随机控制背景，即对布朗运动下的粒子施加控制。特别地，扩散项的存在对确定性最优控制中传统的张量网络方法提出了新的挑战。为克服此困难，我们采用一种基于回归的通用公式，其损失项由 Bellman 一致性误差与一个 Sobolev 型惩罚项组合而成。我们提出了两种新颖的基于草图技术的子程序，用于获取动作-价值函数与价值函数的张量网络近似，这极大地加速了后续回归阶段的收敛速度。我们成功地将所提方法应用于两个具有挑战性的控制问题，即包含 Ginzburg-Landau 势的 1 维和 2 维（共 64 个变量）问题。