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Data-driven approach for extracting the most probable exit trajectory of stochastic dynamical systems with non-Gaussian Lévy noise

提取受非高斯列维噪声扰动随机动力系统的最大似然转移路径 的数据驱动方法

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Abstract

The burgeoning data-driven techniques endow large potential to predict fairly practical or complex dynamical systems in various fields through massive data. Lévy noise, a more universal and intricate fluctuation model comparing with Gaussian white noise, is widely employed in many non-Gaussian cases to mimic bursting or hopping. In this manuscript, we present a systematic data-driven method to identify the most probable exit trajectory of a system that is perturbed both by Gaussian white noise and non-Gaussian Lévy noise. The main theoretical and numerical conceptions involve a set of extended Kramers-Moyal formulas and the Kolmogorov forward equation in classic dynamical systems theory as well as a supervise learning theory to solve the fitting problems by using the Cross Validation. We then give two examples to show the feasibility in detail, and do a brief bifurcation analysis for the most probable exit trajectory. The above approach will serve as a numerical correspondence to as well as verification for the relative theoretical research, and provide a referential resolution to the numerical identification of more transition indicators of this complex system, which is more general than the Gaussian diffusion process.

摘要

机器学习和数据科学技术的飞速发展, 很大程度上满足了许多领域通过数据预测比较实际或复杂的动力系统的需求. 列维噪 声是一个比高斯白噪声更普适和复杂的涨落模型, 它被广泛应用于许多非高斯情形来模拟爆炸或跳跃行为. 在本文中, 我们设计了一 个系统的数据驱动方法来识别受高斯白噪声和非高斯列维噪声扰动的系统的最大似然转移路径. 其中涉及的主要理论和数值概念包 括经典动力系统理论中的非局部Kramers-Moyal公式、非局部Fokker-Planck方程以及一个通过交叉验证解决稀疏回归问题的机器学习 框架. 接下来我们给出两个例子从细节方面展现了该方法的可操作性, 并且对最大似然转移路径做了简要的分岔分析. 该方法将作为 相关理论研究对应的数值方法以及数值角度的验证, 并为该高斯扩散过程的推广复杂系统的其他诸如平均离出时间或离出概率等动 力学指征的数值识别工作提供一些洞见.

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Acknowledgements This work was supported by the National Natural Science Foundation of China (Grant No. 12172167) and A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

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Authors and Affiliations

  1. State Key Laboratory of Mechanics and Control for Mechanical Structures, College of Aerospace Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Linghongzhi Lu  (陆凌弘志) & Xianbin Liu  (刘先斌)

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

    Yang Li  (李扬)

Authors
  1. Linghongzhi Lu  (陆凌弘志)
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  2. Yang Li  (李扬)
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  3. Xianbin Liu  (刘先斌)
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Contributions

Author contributions Linghongzhi Lu contributed to the conceptualization, data curation, formal analysis, investigation, methodology, software, validation, writing of the original draft, and writing-review and editing. Yang Li contributed to the conceptualization, methodology, formal analysis, writing-review and supervision. Xianbin Liu contributed to writing-review and supervision, funding acquisition.

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Correspondence to Xianbin Liu  (刘先斌).

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Conflict of interest On behalf of all authors, the corresponding author states that there is no conflict of interest.

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Lu, L., Li, Y. & Liu, X. Data-driven approach for extracting the most probable exit trajectory of stochastic dynamical systems with non-Gaussian Lévy noise. Acta Mech. Sin. 40, 523094 (2024). https://doi.org/10.1007/s10409-023-23094-x

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