In this paper, we propose a general framework for solving high-dimensional partial differential equations with tensor networks. Our approach offers a comprehensive solution methodology, wherein we employ a combination of particle simulations to update the solution and re-estimations of the new solution as a tensor-network using a recently proposed tensor train sketching technique. Our method can also be interpreted as an alternative approach for performing particle number control by assuming the particles originate from an underlying tensor network. We demonstrate 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.

翻译：本文提出一个通用框架，用于求解高维偏微分方程中的张量网络问题。该方法提供了一套完整的求解策略，其中我们结合粒子模拟更新解，并利用近期提出的张量训练草图技术将新解重构为张量网络形式。该方法亦可解释为一种通过假设粒子源于底层张量网络实现粒子数控制的替代方案。我们通过两个具体场景验证了该方法的普适性与灵活性：基于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。