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指数增长的方法。本文通过直接逼近马尔可夫算子，引入一种张量网络方法求解Kolmogorov后向方程。我们证明，借助层次化草图算法，高维马尔可夫算子可在函数层次张量假设下获得。当终端条件满足函数层次张量假设时，所提出的算子通过高效的函数张量网络收缩过程输出偏微分方程解的函数层次张量表示。此外，当初始分布具有函数层次张量形式时，所提出的基于算子的方法也为求解Kolmogorov前向方程提供了高效途径。我们成功将所提方法应用于两个具有数百个变量的含时Ginzburg-Landau挑战性模型。