In this paper, we propose a general framework for solving high-dimensional partial differential equations with tensor networks. Our approach uses a 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：本文提出了一种通用的张量网络框架，用于求解高维偏微分方程。该方法通过蒙特卡洛模拟更新数值解，并利用近期提出的张量列草图技术从样本数据中重新估计张量网络形式的解。我们通过两个具体场景验证该方法的普适性与灵活性：基于朗之万动力学的福克-普朗克方程模拟，以及基于辅助场量子蒙特卡洛的量子虚时演化。同时提供收敛性保证与数值实验，以证明所提方法的有效性。