Wei Wei, Gao Ting, Chen Xiaoli, Duan Jinqiao
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei 430074, China.
Department of Mathematics, National University of Singapore, Singapore 119077, Singapore.
Chaos. 2022 May;32(5):051102. doi: 10.1063/5.0093924.
Many complex real world phenomena exhibit abrupt, intermittent, or jumping behaviors, which are more suitable to be described by stochastic differential equations under non-Gaussian Lévy noise. Among these complex phenomena, the most likely transition paths between metastable states are important since these rare events may have a high impact in certain scenarios. Based on the large deviation principle, the most likely transition path could be treated as the minimizer of the rate function upon paths that connect two points. One of the challenges to calculate the most likely transition path for stochastic dynamical systems under non-Gaussian Lévy noise is that the associated rate function cannot be explicitly expressed by paths. For this reason, we formulate an optimal control problem to obtain the optimal state as the most likely transition path. We then develop a neural network method to solve this issue. Several experiments are investigated for both Gaussian and non-Gaussian cases.
许多复杂的现实世界现象表现出突然、间歇或跳跃行为，在非高斯 Lévy 噪声下，这些行为更适合用随机微分方程来描述。在这些复杂现象中，亚稳态之间最可能的转变路径很重要，因为这些罕见事件在某些情况下可能会产生很大影响。基于大偏差原理，最可能的转变路径可以被视为连接两点的路径上速率函数的最小值。计算非高斯 Lévy 噪声下随机动力系统最可能转变路径的挑战之一是，相关的速率函数不能通过路径明确表示。因此，我们制定了一个最优控制问题，以获得作为最可能转变路径的最优状态。然后，我们开发了一种神经网络方法来解决这个问题。针对高斯和非高斯情况都进行了几项实验。
