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A machine learning method for computing quasi-potential of stochastic dynamical systems

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Abstract

The concept of quasi-potential plays a central role in understanding the mechanisms of rare events and characterizing the statistics of transition behaviors in stochastic dynamics. Despite its significance, the computation of quasi-potential is a challenging problem with limited existing techniques. In this paper, we devise a machine learning method to compute the quasi-potential of stochastic dynamical systems. More specifically, we first derive the Hamilton-Jacobi equation satisfied by quasi-potential via WKB approximation for two important classes of stochastic processes: diffusion processes and continuous-time jump Markov processes. Then we design a neural network with automatic differentiation technique to compute the quasi-potential based on the Hamilton-Jacobi equation. Two prototypical examples are presented to demonstrate the efficacy and accuracy of our method for stochastic differential equations with additive and multiplicative Gaussian noise and continuous-time jump Markov processes. This approach provides an effective tool in exploring the internal mechanisms of rare events triggered by random perturbations in various scientific fields.

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Acknowledgements

We would like to thank Xiaoli Chen for helpful discussions.

Funding

This work was supported by the NSFC 62073166, 61673215, the 333 Project, and the Key Laboratory of Jiangsu Province.

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Authors and Affiliations

  1. School of Automation, Nanjing University of Science and Technology, Nanjing, 210094, China

    Yang Li & Shengyuan Xu

  2. Departments of Applied Mathematics & Physics, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Jinqiao Duan

  3. State Key Laboratory of Mechanics and Control of Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, 29 Yudao Street, Nanjing, 210016, China

    Xianbin Liu

  4. School of Science, Huzhou University, Huzhou, 313000, China

    Yuming Chu

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  1. Yang Li
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  2. Shengyuan Xu
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  3. Jinqiao Duan
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  4. Xianbin Liu
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  5. Yuming Chu
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Correspondence to Shengyuan Xu.

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Li, Y., Xu, S., Duan, J. et al. A machine learning method for computing quasi-potential of stochastic dynamical systems. Nonlinear Dyn 109, 1877–1886 (2022). https://doi.org/10.1007/s11071-022-07536-x

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