A hierarchical dynamical low-rank algorithm for the stochastic description of large reaction networks

TL;DR

This paper introduces a hierarchical dynamical low-rank algorithm using binary tree tensor networks to efficiently approximate the chemical master equation, significantly reducing memory and computational costs for large reaction networks.

Contribution

It presents a novel hierarchical low-rank approximation method for the CME that partitions reaction networks to improve efficiency and accuracy over traditional Monte Carlo approaches.

Findings

Reduces memory usage in large reaction networks

Improves computational performance over Monte Carlo methods

Achieves better accuracy in numerical examples

Abstract

The stochastic description of chemical reaction networks with the kinetic chemical master equation (CME) is important for studying biological cells, but it suffers from the curse of dimensionality: The amount of data to be stored grows exponentially with the number of chemical species and thus exceeds the capacity of common computational devices for realistic problems. Therefore, time-dependent model order reduction techniques such as the dynamical low-rank approximation are desirable. In this paper we propose a dynamical low-rank algorithm for the kinetic CME using binary tree tensor networks. The dimensionality of the problem is reduced in this approach by hierarchically dividing the reaction network into partitions. Only reactions that cross partitions are subject to an approximation error. We demonstrate by two numerical examples (a 5-dimensional lambda phage model and a 20-dimensional reaction cascade) that the proposed method drastically reduces memory consumption and shows improved computational performance and better accuracy compared to a Monte Carlo method.