Extracting stochastic dynamical systems with α-stable Levy noise from data

With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extract stochastic dynamical systems with (non-Gaussian) Levy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with alpha-stable Levy noise from sample path data based on the properties of alpha-stable distributions. More specifically, we first estimate the Levy jump measure and noise intensity via computing mean and variance of the amplitude of the increment of the sample paths. Then we approximate the drift coefficient by combining nonlocal Kramers-Moyal formulas with normalizing flows. Numerical experiments on one- and two-dimensional prototypical examples including simulated and real world measurement data illustrate the accuracy and effectiveness of our method. This approach will become an effective scientific tool in discovering stochastic governing laws of complex phenomena and understanding dynamical behaviors under non-Gaussian fluctuations.