Detecting physical laws from data of stochastic dynamical systems perturbed by non-Gaussian $\alpha$-stable Lévy noise

Abstract

Abstract Massive data from observations, experiments and simulations of dynamical models in scientific and engineering fields make it desirable for data-driven methods to extract basic laws of these models. We present a novel method to identify such high dimensional stochastic dynamical systems that perturbed by a non-Gaussian $\alpha$-stable Lévy noise. More explicitly, firstly a machine learning framework to solve the sparse regression problem is established to grasp the drift terms through one of non-local Kramers-Moyal formulas. Then the jump measure and intensity of the noise are disposed by the relationship with statistical characteristics of the process. Three examples are then given to demonstrate the feasibility. This approach proposes an effective way to understand the complex phenomena of systems under non-Gaussian fluctuations and illuminates some insights into the exploration for further typical dynamical indicators such as the maximum likelihood trasition path or mean exit time of these stochastic systems.