Neural-network solutions to stochastic reaction networks

Abstract

The stochastic reaction network is widely used to model stochastic processes in physics, chemistry and biology. However, the size of the state space increases exponentially with the number of species, making it challenging to investigate the time evolution of the chemical master equation for the reaction network. Here, we propose a machine-learning approach using the variational autoregressive network to solve the chemical master equation. The approach is based on the reinforcement learning framework and does not require any data simulated in prior by another method. Different from simulating single trajectories, the proposed approach tracks the time evolution of the joint probability distribution in the state space of species counts, and supports direct sampling of configurations and computing their normalized joint probabilities. We apply the approach to various systems in physics and biology, and demonstrate that it accurately generates the probability distribution over time in the genetic toggle switch, the early life self-replicator, the epidemic model and the intracellular signaling cascade. The variational autoregressive network exhibits a plasticity in representing the multimodal distribution by feedback regulations, cooperates with the conservation law, enables time-dependent reaction rates, and is efficient for high-dimensional reaction networks with allowing a flexible upper count limit. The results suggest a general approach to investigate stochastic reaction networks based on modern machine learning.

Learning nonequilibrium statistical mechanics and dynamical phase transitions

“…The present approach is applicable to other types of Markovian dynamics, including stochastic reaction networks 26,50 , where phase transitions can be analyzed after adding the control parameter. Based on generality of the VAN, it is adaptable to other topologies, such as the voter model on graphs 51 and epidemic spreading on networks 52 , where architecture of the VAN can be the graph neural network 43 .…”

“…( 2)). The renormalization procedure over time points enables to extraction of the dynamical partition function under the tilted generator, beyond the algorithm of only tracking the evolving distribution under the non-tilted generator 26 .…”

“…In this work, we develop a general framework to track nonequilibrium systems over time by variational autoregressive networks (VAN), which offers an ideal model for describing the normalized joint distribution of configurations 19 . The VAN was previously shown effective to investigate equilibrium statistical mechanics [20][21][22] , quantum many-body systems [23][24][25] , chemical reaction networks 26 and computational biology 27 . However, evaluating the dynamical partition function and characterizing dynamical phase transitions have not been achieved in these applications of the VAN.…”

A divide-and-conquer method for analyzing high-dimensional noisy gene expression networks

“…To tackle more complex systems, our method needs further development to incorporate additional biological mechanisms such as time delays, multiscale phenomena, and cell-to-cell interactions. Additionally, it is worth exploring ways to enhance the performance of the RB-CME solver, such as incorporating deep learning methods (45)(46)(47) to our divide-and-conquer approach.…”

“…Similarly, the moment closure method closes the moment equations based on a chosen family of distributions and then tracks the first few moments (37)(38)(39)(40)(41)(42)(43)(44). More recently, some deep learning approaches (45)(46)(47) were established for solving CMEs thanks to the universal approximation properties of neural networks. These parametric methods are quite successful in many applications.…”