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：本文提出了一种通用框架，用于求解高维偏微分方程的张量网络方法。我们的方案提供了一套完整的解决途径，其中结合了粒子模拟更新解与基于最新张量列摄动技术的张量网络重估计。该方法也可视为一种替代性的粒子数控制手段，其核心假设源于粒子由底层张量网络生成。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。