Solving high-dimensional partial differential equations necessitates methods free of exponential scaling in the dimension of the problem. This work introduces a tensor network approach for the Kolmogorov backward equation via approximating directly the Markov operator. We show that the high-dimensional Markov operator can be obtained under a functional hierarchical tensor (FHT) ansatz with a hierarchical sketching algorithm. When the terminal condition admits an FHT ansatz, the proposed operator outputs an FHT ansatz for the PDE solution through an efficient functional tensor network contraction procedure. In addition, the proposed operator-based approach also provides an efficient way to solve the Kolmogorov forward equation when the initial distribution is in an FHT ansatz. We apply the proposed approach successfully to two challenging time-dependent Ginzburg-Landau models with hundreds of variables.

翻译：求解高维偏微分方程需要避免问题维度呈指数增长的方法。本文提出一种张量网络方法，通过直接逼近马尔可夫算子来求解Kolmogorov后向方程。我们证明，利用层次化草图算法，可在功能化层次张量（FHT）假设下获得高维马尔可夫算子。当终端条件满足FHT假设时，所提出的算子可通过高效的功能化张量网络缩并过程输出偏微分方程解的FHT形式。此外，当初始分布同样满足FHT假设时，该基于算子的方法也为求解Kolmogorov前向方程提供了高效途径。我们成功将该方法应用于包含数百个变量的两个含时Ginzburg-Landau模型。