Solving high-dimensional Hamilton-Jacobi-Bellman equation with functional hierarchical tensor
<|reference_start|>Solving high-dimensional Hamilton-Jacobi-Bellman equation with functional hierarchical tensor: 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.<|reference_end|>
@article{tang2024solving,
    title={Solving high-dimensional Hamilton-Jacobi-Bellman equation with
  functional hierarchical tensor},
    author={Xun Tang, Nan Sheng and Lexing Ying},
    journal={arXiv preprint arXiv:2408.04209},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.04209},
    primaryClass={math.NA cs.NA}
}