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Machine learning framework for computing the most probable paths of stochastic dynamical systems

Yang Li, Jinqiao Duan, and Xianbin Liu
Phys. Rev. E 103, 012124 – Published 21 January 2021
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  • INTRODUCTION
  • ONSAGER-MACHLUP THEORY
  • NUMERICAL ALGORITHMS
  • NUMERICAL EXPERIMENTS
  • DISCUSSION
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    The emergence of transition phenomena between metastable states induced by noise plays a fundamental role in a broad range of nonlinear systems. The computation of the most probable paths is a key issue to understanding the mechanism of transition behaviors. The shooting method is a common technique for this purpose to solve the Euler-Lagrange equation for the associated action functional, while losing its efficacy in high-dimensional systems. In the present work, we develop a machine learning framework to compute the most probable paths in the sense of Onsager-Machlup action functional theory. Specifically, we reformulate the boundary value problem of a Hamiltonian system and design a neural network to remedy the shortcomings of the shooting method. The successful applications of our algorithms to several prototypical examples demonstrate its efficacy and accuracy for stochastic systems with both (Gaussian) Brownian noise and (non-Gaussian) Lévy noise. This approach is effective in exploring the internal mechanisms of rare events triggered by random fluctuations in various scientific fields.

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    • Received 6 October 2020
    • Revised 24 December 2020
    • Accepted 8 January 2021

    DOI:https://doi.org/10.1103/PhysRevE.103.012124

    ©2021 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Biological neural networksDiffusionFluctuations & noise
    Statistical Physics & ThermodynamicsNonlinear DynamicsNetworks

    Authors & Affiliations

    Yang Li1,2,*, Jinqiao Duan2,†, and Xianbin Liu1,‡

    • 1State 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
    • 2Department of Applied Mathematics, College of Computing, Illinois Institute of Technology, Chicago, Illinois 60616, USA

    • *li_yang@nuaa.edu.cn
    • †duan@iit.edu
    • ‡Corresponding author: xbliu@nuaa.edu.cn

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    Issue

    Vol. 103, Iss. 1 — January 2021

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    Images

    • Figure 1
      Figure 1

      Architecture of neural network with L hidden layers. xfi and λi, i=1,...,n, are the input and output of the neural network, respectively. aj(l) represents the value of the j th neuron in the l th layer with j=1,...,nl and l=1,...,L.

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    • Figure 2
      Figure 2

      The potential function U(x) of the energy balance model. The two minima correspond to the colder glacial state xs1=233.52K (−39.63∘C) and warmer interstadial state xs2=288.03K (14.88∘C), separated by one unstable state xu.

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    • Figure 3
      Figure 3

      M=50 randomly sampled points to generate input and output data of the neural network. (a) Integrated end points xf. (b) Randomly sampled λ.

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    • Figure 4
      Figure 4

      The values of the cost function of the neural network during the training process in the energy balance model.

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    • Figure 5
      Figure 5

      Comparison of most probable paths between using a neural network and with a randomly sampled λ. The red dotted curve denotes the most probable path using a neural network, and other curves indicate the paths with random λ. The pentagram denotes a warmer state xs2=288.03K.

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    • Figure 6
      Figure 6

      Comparison of the most probable transition paths and conjugate momenta between being computed by a neural network (denoted as a blue line), directly integrated by a Hamiltonian system from initial point (x(0),p(0)) (indicated as a black plus sign), and computed by a minimum action method (shown as a red circle). (a) The most probable transition paths. (b) Conjugate momenta.

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    • Figure 7
      Figure 7

      M=1000 randomly sampled λ and corresponding integrated end points xf for Case 1 to generate input and output data of the neural network. (a) xf2 vs xf1; (b) xf3 vs xf1; (c) λ2 vs λ1; (d) λ3 vs λ1.

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    • Figure 8
      Figure 8

      The values of the cost function of a neural network during the training process for Case 1 in the Lorenz model.

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    • Figure 9
      Figure 9

      Comparison of the first component x1(t) of the most probable paths between using a neural network and with a randomly sampled λ for Case 1. The red dotted curve denotes the most probable path using a neural network, and other curves indicate the paths with a randomly sampled λ. The pentagram denotes the target value 1.

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    • Figure 10
      Figure 10

      Comparison of the most probable transition paths and conjugate momenta between a neural network (denoted as lines) and the minimum action method (indicated as circle, cross, and square sign) for Case 1. (a) The most probable transition paths. (b) Conjugate momenta.

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    • Figure 11
      Figure 11

      Comparison of the most probable transition paths between using a neural network (denoted as lines) and with the minimum action method (indicated as circle, cross, and square sign) for Case 2 with (a) β1=−0.5 and (b) β1=0.5.

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    • Figure 12
      Figure 12

      Comparison of the most probable transition paths between using a neural network (denoted as lines) and with a minimum action method (indicated as circle, cross, and square sign) for Case 3 with (a) μ=1 and (b) μ=2.

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    • Figure 13
      Figure 13

      Comparison of the most probable transition paths between using a neural network (denoted as lines) and with a minimum action method (indicated as circle, cross, and square sign) for Case 4 with (a) κ=0 and (b) κ=1.

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