Committor functions via tensor networks

TL;DR

This paper introduces a tensor network-based method to efficiently compute committor functions in high-dimensional stochastic systems, overcoming traditional computational limitations.

Contribution

It presents a novel tensor train parametrization of committor functions and equilibrium densities, enabling scalable solutions for high-dimensional problems.

Findings

Effective in high-dimensional settings

Reduces computational complexity to linear in dimensions

Avoids sampling from complex equilibrium distributions

Abstract

We propose a novel approach for computing committor functions, which describe transitions of a stochastic process between metastable states. The committor function satisfies a backward Kolmogorov equation, and in typical high-dimensional settings of interest, it is intractable to compute and store the solution with traditional numerical methods. By parametrizing the committor function in a matrix product state/tensor train format and using a similar representation for the equilibrium probability density, we solve the variational formulation of the backward Kolmogorov equation with linear time and memory complexity in the number of dimensions. This approach bypasses the need for sampling the equilibrium distribution, which can be difficult when the distribution has multiple modes. Numerical results demonstrate the effectiveness of the proposed method for high-dimensional problems.