Data-driven method to extract mean exit time and escape probability for dynamical systems driven by Lévy noises

Abstract

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 [Formula: see text]-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.