Complex dynamical systems have been investigated through many data-driven methods with easily accessible and massive data from observations, experiments or simulations in recent decades. However, few works dealt with the stochastical non-Gaussian perturbation case. In this paper, for a class of systems perturbed by non-Gaussian $\alpha$-stable Lévy noises, we devise a data-driven approach to extract the mean exit time and escape probability of rare transition dynamics. The theories are based on the non-local Kramers–Moyal formulas and non-local partial differential equations generated by Kolmogorov backward operator, accompanied with the corresponding numerical algorithms. Specifically, we first identify governing laws by a machine learning framework according to non-local Kramers–Moyal formulas. With learned systems, the mean exit time and escape probability are obtained by solving corresponding partial differential equations. The feasibility and accuracy of the method are checked by one- and two-dimensional examples. This method will serve as an example to study stochastic systems with non-Gaussian perturbations from data and illuminate some insights into the extraction of other dynamical indicators like the maximum likelihood transition path.

近几十年来，人们通过许多数据驱动的方法对复杂的动力系统进行了研究，这些方法利用来自观察、实验或模拟的易于访问的大量数据。然而，很少有研究涉及随机非高斯扰动情况。本文针对一类受非高斯扰动的系统$\alpha$-稳定的 Lévy 噪声，我们设计了一种数据驱动的方法来提取罕见过渡动态的平均退出时间和逃逸概率。该理论基于Kolmogorov后向算子生成的非局部Kramers-Moyal公式和非局部偏微分方程，并配有相应的数值算法。具体来说，我们首先根据非局部 Kramers-Moyal 公式通过机器学习框架确定管辖法律。对于学习系统，平均退出时间和逃逸概率是通过求解相应的偏微分方程获得的。通过一维和二维算例验证了该方法的可行性和准确性。该方法将作为研究具有数据非高斯扰动的随机系统的示例，并阐明提取其他动态指标（例如最大似然转移路径）的一些见解。