An adaptive solution to the chemical master equation using quantized tensor trains with sliding windows

Abstract

To cope with an extremely large or even infinite state space when solving the chemical master equation in biological problems, a potent strategy is to restrict to a finite state projection (FSP) and represent the transition matrix and probability vector in quantized tensor train (QTT) format, leading to savings in storage while retaining accuracy. In an earlier adaptive FSP–QTT algorithm, the multidimensional state space was downsized and kept in the form of a hyper rectangle that was updated when needed by selectively doubling some of its side dimensions. However, this could result in a much larger state space than necessary, with the effect of hampering both the execution time and stepping scheme. In this work, we improve the algorithm by enabling sliding windows that can dynamically slide, shrink or expand, with updates driven by a number of stochastic simulation algorithm trajectories. The ensuing state space is a considerably reduced hyper rectangle containing only the most probable states at each time step. Three numerical experiments of varying difficulty are performed to compare our approach with the original adaptive FSP–QTT algorithm.

Approximating solutions of the Chemical Master equation using neural networks

“…The CME cannot be solved analytically for most biologically relevant cases, and as the state space is typically infinite, numerical solutions to the CME often involve state space truncation methods such as the Finite State Projection (FSP) (Munsky and Khammash, 2006). However, owing to the combinatorial explosion of the state space in the number of species, using the FSP to solve the CME quickly becomes too computationally intensive for most non-trivial systems (Kazeev et al, 2014;Kazeev and Schwab, 2015;Dinh and Sidje, 2020). A wide variety of other approximation methods exist for the CME (see Schnoerr et al (2017) for an overview), but these often trade computational efficiency for accuracy and are generally difficult to apply to complex systems involving many species and interactions.…”

Approximating Solutions of the Chemical Master Equation using Neural Networks

“…The CME cannot be solved analytically for most biologically relevant cases, and since the state space is typically infinite, numerical solutions of the CME often involve state space truncation methods such as the Finite State Projection (FSP) [7]. However, due to the combinatorial explosion of the state space in the number of species, using the FSP to solve the CME quickly becomes too computationally intensive for most non-trivial systems [8][9][10]. A wide variety of other approximation methods exist for the CME (see [6] for an overview), but these often trade computational efficiency for accuracy and are generally difficult to apply to complex systems involving many species and interactions.…”

Inference and uncertainty quantification of stochastic gene expression via synthetic models

“…Among the most well known of these are the finite-state projection (FSP) [ 11 ], continuum approximations [ 7 , 12 ] and moment equations [ 13 ]. The FSP solves the CME on a finite truncation of the state space, whose size typically grows exponentially in the number of species; in practice, this approach relies on computationally intensive approximations [ 14 – 16 ] for more complex systems. Continuum approximations to the CME based on stochastic differential equations, such as the chemical Langevin formalism [ 12 ] and the linear noise approximation (LNA) [ 7 ] are limited to systems with small noise and in the case of the latter, Gaussian copy number distributions.…”