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Neural-network solutions to stochastic reaction networks

A preprint version of the article is available at arXiv.


The stochastic reaction network in which chemical species evolve through a set of reactions is widely used to model stochastic processes in physics, chemistry and biology. To characterize the evolving joint probability distribution in the state space of species counts requires solving a system of ordinary differential equations, the chemical master equation, where the size of the counting state space increases exponentially with the type of species. This makes it challenging to investigate the stochastic reaction network. Here we propose a machine learning approach using a variational autoregressive network to solve the chemical master equation. Training the autoregressive network employs the policy gradient algorithm in the reinforcement learning framework, which does not require any data simulated previously by another method. In contrast with simulating single trajectories, this approach tracks the time evolution of the joint probability distribution, and supports direct sampling of configurations and computing their normalized joint probabilities. We apply the approach to representative examples in physics and biology, and demonstrate that it accurately generates the probability distribution over time. The variational autoregressive network exhibits plasticity in representing the multimodal distribution, cooperates with the conservation law, enables time-dependent reaction rates and is efficient for high-dimensional reaction networks, allowing a flexible upper count limit. The results suggest a general approach to study stochastic reaction networks based on modern machine learning.

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Fig. 1: Tracking the joint probability distribution of stochastic reaction networks over time.
Fig. 2: The result of the genetic toggle switch.
Fig. 3: The result of the linear signalling cascade.

Data availability

The authors declare that the data supporting this study are available within the paper. Source data are provided with this paper.

Code availability

A PyTorch implementation of the algorithm can be found at GitHub (, Code Ocean ( and Zenodo (


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We thank J. Liu for sharing the code of the transformer. We acknowledge J. Liang, M. Khammash, A. Gupta, A. Hoffmann, A. Farhat, F. Manuchehrfar and members of Online Club Nanothermodynamica for helpful discussions. This work is supported by projects 12105014 (Y.T.), 11747601 (P.Z.) and 11975294 (P.Z.) of the National Natural Science Foundation of China. P.Z. acknowledges the WIUCASICTP2022 grant. The high-performance computing is supported by Dawning Information Industry and the Interdisciplinary Intelligence SuperComputer Center of Beijing Normal University, Zhuhai.

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Authors and Affiliations



Y.T. and P.Z. had the original idea for this work, Y.T. and J.W. performed the study. All authors contributed to the preparation of the manuscript.

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Correspondence to Ying Tang or Pan Zhang.

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The authors declare no competing interests.

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Nature Machine Intelligence thanks Nigel Goldenfeld and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Supplementary Notes 1–2, Figs. 1–13 and Tables 1–2.

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Tang, Y., Weng, J. & Zhang, P. Neural-network solutions to stochastic reaction networks. Nat Mach Intell 5, 376–385 (2023).

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