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基于大偏差理论非高斯随机动力系统离出行为研究

李扬 赵锋 刘先斌

李扬, 赵锋, 刘先斌. 基于大偏差理论非高斯随机动力系统离出行为研究. 力学进展, 待出版 doi: 10.6052/1000-0992-21-033
引用本文: 李扬, 赵锋, 刘先斌. 基于大偏差理论非高斯随机动力系统离出行为研究. 力学进展, 待出版 doi: 10.6052/1000-0992-21-033
Li Y, Zhao F, Liu X B. On the exit behaviors of non-Gaussian stochastic dynamical systems based on large deviation theory. Advances in Mechanics, in press doi: 10.6052/1000-0992-21-033
Citation: Li Y, Zhao F, Liu X B. On the exit behaviors of non-Gaussian stochastic dynamical systems based on large deviation theory. Advances in Mechanics, in press doi: 10.6052/1000-0992-21-033

基于大偏差理论非高斯随机动力系统离出行为研究——

doi: 10.6052/1000-0992-21-033
基金项目: 国家自然科学基金资助项目 (11772149).
详细信息
    作者简介:

    李扬, 男, 1991年出生, 南京理工大学自动化学院紫金博士后. 本科毕业于南京航空航天大学工程力学“钱伟长班”, 博士毕业于南京航空航天大学一般力学与力学基础专业, 博士期间公派留学到美国伊利诺伊理工大学应用数学系跟随Jinqiao Duan教授做研究一年. 博士期间主要从事非线性随机动力学和数据科学方向的研究. 目前已在《Physica D》、《Physical Review E》、《Chaos》、《Communications in Nonlinear Science and Numerical Simulation》、《力学学报》等国内外学术期刊上发表论文9篇, 其中SCI论文8篇 (第一作者5篇). 主要研究兴趣包括: 大偏差理论、机器学习、数据科学、离出问题、非高斯噪声、随机混合系统等

    刘先斌, 男, 1965年出生, 南京航空航天大学航空学院教授, 博士生导师. 主要从事非线性随机动力学: 局部和全局随机分岔、噪声诱发混沌系统随机动力学行为、随机颤振、离出问题和随机共振等方面的研究. 目前主持国家自然科学基金面上项目2项. 已主持完成国家自然科学基金项目3项、教育部博士点基金和国家留学归国基金各1项、国家重点实验室基金项目2项. 此外, 作为主要研究人员, 参与完成国家自然科学基金重点项目、重大研究计划和面上项目4项. 已发表学术论文80余篇、论著1部. 论文发表在: 《Proceedings of the Royal Society London A》、《ASME-Journal of Applied Mechanics》、《Physical Review E》、《Chaos》、《Nonlinear Dynamics》、《力学学报》、《中国科学》等国际、国内学术期刊, 其中SCI检索期刊论文70余篇, 论文论著他引500篇次

    通讯作者:

    xbliu@nuaa.edu.cn

  • 中图分类号: O324

On the exit behaviors of non-Gaussian stochastic dynamical systems based on large deviation theory

More Information
  • 摘要: 本文介绍了大偏差理论的基本思想及其在非高斯随机动力系统的离出问题研究中的应用. 依据不同的非高斯噪声类型, 本文分别评述了随机混合系统、指数轻跳跃过程和$\alpha $-稳定Lévy噪声驱动的随机动力系统的离出问题的主要研究方法和近期研究进展. 针对随机混合系统, 本文介绍了利用随机微分方程对其进行近似的拟稳态扩散近似方法, 计算拟势和最优离出路径的WKB近似方法, 和细致平衡条件的研究, 以及求解随机混合系统的简化版本 (即生灭过程) 的离出问题的研究进展. 对于指数轻跳跃过程驱动的随机动力系统, 本文介绍了其大偏差原理和中度偏差原理的泛函极值问题的建立, 拟势概念的定义和平均离出时间的估计. 针对具有$\alpha $-稳定Lévy噪声的随机动力系统, 本文介绍了计算平均首次离出时间和离出概率的理论和数值方法, 计算最优离出路径的Onsager-Machlup理论、机器学习方法、最大似然法和数据驱动方法. 最后, 给出了非高斯随机动力系统的离出现象相关的一些开放性问题.

     

  • 图  1  平均场方程关于$\theta $从单稳态到双稳态的分岔图. $\theta = 0.95$时, 存在两个稳定不动点${x_1}$, ${x_2}$和一个不稳定不动点${x_0}$

    图  2  哈密顿系统的相图和离出路径. 粉色和亮蓝曲线分别代表从${x_1}$${x_2}$${x_2}$${x_1}$的离出路径. 绿色和棕色虚线表示扩散近似的离出路径

    图  3  (a) 扩散近似和原始系统的拟势对比. (b) 平均离出时间的对数和噪声强度的倒数之间的关系

    图  4  细致平衡条件分析示意图

    图  5  具有$L$层隐藏层的神经网络的结构. ${x_{fi}}$${\lambda _i}$, $i = 1,\; \cdots ,\;n$分别是神经网络的输入和输出. $a_j^{\left( l \right)}$表示第$l$层第$j$个神经元的值, 其中$j = 1,\; \cdots ,\;{n_l}$, $l = 1,\;\cdot,\;L$

    表  1  几种典型随机过程的细致平衡条件

    随机过程细致平衡条件文献
    扩散过程加性高斯、
    有势系统
    满足细致平衡朱位秋 1992
    一般情况$\begin{array}{r} {b_i}\left( {\bf{x} } \right){p_s}\left( {\bf{x} } \right) + {\varepsilon _i}{b_i}\left( {\varepsilon {\bf{x} } } \right){p_s}\left( {\bf{x} } \right) - \displaystyle\sum\limits_j {\frac{\partial }{ {\partial {x_j} } }\left[ { {a_{ij} }\left( {\bf{x} } \right){p_s}\left( {\bf{x} } \right)} \right]} = 0 \\ {\varepsilon _i}{\varepsilon _j}{a_{ij} }\left( {\varepsilon{\bf{ x} } } \right) - {a_{ij} }\left( {\bf{x} } \right) = 0 \end{array}$
    ($b$为漂移系数, $a$为扩散矩阵, ${p_s}$为平稳概率分布,
    ${\varepsilon _i} = \pm 1$依赖于变量奇偶性)
    跳跃Markov过程${W_{ji}}{P_i} = {W_{ij}}{P_j}$
    (${W_{ij}}$为转移率, ${P_i}$为平稳概率)
    Dykman et al. 1994
    随机混合系统不满足细致平衡Li & Liu 2019
    下载: 导出CSV
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  • 收稿日期:  2021-06-10
  • 录用日期:  2021-09-06
  • 网络出版日期:  2021-09-17

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