Solving high-dimensional Kolmogorov backward equations with functional hierarchical tensor operators

TL;DR

This paper addresses solving high-dimensional Kolmogorov backward equations by directly approximating the Markov operator with a functional hierarchical tensor ($FHT$) and a hierarchical sketching scheme to represent the joint density of $(X_0,X_t)$. By leveraging Bayes’ rule, the method reduces the problem to estimating a density operator that can be contracted with an $FHT$-amenable terminal condition to obtain the solution, with extensions to the Kolmogorov forward equation when the initial distribution is $FHT$-structured. The key contributions include introducing the $FHT$ operator representation, detailing a hierarchical sketching procedure for joint densities, and providing a concrete implementation for time-dependent Ginzburg-Landau models in 1D and 2D that demonstrates accurate solutions with hundreds of dimensions. This framework offers scalable, tensor-network-based tools for high-dimensional stochastic PDEs, with potential impact in physics-informed modeling and beyond. Open questions include extending to broader nonlinear PDEs such as Hamilton-Jacobi and exploring further efficiency gains in basis design and sketching strategies.

Abstract

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.

Figures (10)

• Figure 1: Binary decomposition of variables
• Figure 1: Illustration of the tensor contraction diagram for the approximated solution to the Kolmogorov backward equation for $d = 8$.
• Figure 1: 1D Ginzburg-Landau model. The plots of the marginal distribution at $(x_{15}, x_{25})$ and $(y_{15}, y_{25})$.
• Figure 2: 1D Ginzburg Landau model. The plot compares the function evaluation of the FHT-based approximation with reference values obtained from Monte-Carlo estimations, where the error bar indicates the standard deviation from each Monte-Carlo estimation.
• Figure 3: 1D Ginzburg Landau model. Plot of $\iota(x(t))$, i.e. the propensity of $x(t) = (t,\ldots, t)$ to enter a proximity of $y_{+} = (1, \ldots, 1)$. One can see that the Markov operator approach can capture the sharp transition at $t = 0$.