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.

翻译：利用动力学化学主方程（CME）对化学反应网络进行随机描述对于研究生物细胞至关重要，但其面临维度灾难：所需存储的数据量随化学物种数量的增加呈指数级增长，因此在处理实际问题时常超出普通计算设备的容量。因此，采用诸如动态低秩近似等时变模型降阶技术是必要的。本文提出一种基于二叉树张量网络的动力学CME动态低秩算法。该方法通过将反应网络层次化划分为多个分区来降低问题的维度。仅跨越分区的反应会引入近似误差。我们通过两个数值算例（一个5维λ噬菌体模型和一个20维反应级联）证明，所提方法能显著降低内存消耗，并在计算性能和精度方面均优于蒙特卡洛方法。