AI 中文总结
该研究提出适用于任意拓扑张量网络的基于SVD的分解方法及可追踪张量图的相关程序，将其用于高维PDE积分，在多元函数分解和Fokker-Planck方程求解中，相比经典张量格式精度相当或更优且自由度少、成本低。
AI 中文摘要
我们提出一种新的基于SVD的张量分解方法，适用于具有任意图拓扑结构的张量网络，将经典的基于分层SVD的技术扩展到具有循环和一般连通性的网络。我们还为可追踪张量图引入了加法和舍入程序，使得高维PDE可以直接以图格式进行分步时间积分，且每一步都能将秩截断控制在规定的容差范围内。我们在多元函数分解和Fokker-Planck方程的数值求解中验证了该新方法，发现图格式表示的精度与经典的张量列（Tensor Train）和分层塔克（Hierarchical Tucker）张量格式相当或更优，同时自由度显著更少，计算成本更低。
英文摘要
We introduce a new SVD-based tensor decomposition method for tensor networks with arbitrary graph topologies, extending classical hierarchical SVD-based techniques to networks with cycles and general connectivity. We also introduce addition and rounding procedures for traceable tensor graphs, enabling step-rounding time integration of high-dimensional PDEs directly in graph format, with rank truncation controlled to a prescribed tolerance at every time step. We demonstrate the new method on the decomposition of multivariate functions and on the numerical solution of the Fokker-Planck equation, and find that the graph-format representation attains comparable or better accuracy than the classical tensor train and hierarchical Tucker tensor formats, while using substantially fewer degrees of freedom at lower computational cost.
Comments24 pages, 15 figures
