In this paper, we propose a general framework for solving high-dimensional partial differential equations with tensor networks. Our approach uses Monte-Carlo simulations to update the solution and re-estimates the new solution from samples as a tensor-network using a recently proposed tensor train sketching technique. We showcase the versatility and flexibility of our approach by applying it to two specific scenarios: simulating the Fokker-Planck equation through Langevin dynamics and quantum imaginary time evolution via auxiliary-field quantum Monte Carlo. We also provide convergence guarantees and numerical experiments to demonstrate the efficacy of the proposed method.

翻译：本文提出一个通用框架，用于通过张量网络求解高维偏微分方程。我们的方法利用蒙特卡洛模拟更新解，并基于样本使用近期提出的张量列草图技术重新将新解估计为张量网络形式。我们通过将方法应用于两个具体场景来展示其通用性与灵活性：通过朗之万动力学模拟福克-普朗克方程，以及通过辅助场量子蒙特卡洛实现量子虚时间演化。我们还提供了收敛性保证与数值实验，以证明所提方法的有效性。