Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems

Abstract: Advances in data science are leading to new progresses in the analysis and understanding of complex dynamics for systems with experimental and observational data. With numerous physical phenomena exhibiting bursting, flights, hopping, and intermittent features, stochastic differential equations with non-Gaussian L\'evy noise are suitable to model these systems. Thus it is desirable and essential to infer such equations from available data to reasonably predict dynamical behaviors. In this work, we consider a data-driven method to extract stochastic dynamical systems with non-Gaussian asymmetric (rather than the symmetric) L\'evy process, as well as Gaussian Brownian motion. We establish a theoretical framework and design a numerical algorithm to compute the asymmetric L\'evy jump measure, drift and diffusion (i.e., nonlocal Kramers-Moyal formulas), hence obtaining the stochastic governing law, from noisy data. Numerical experiments on several prototypical examples confirm the efficacy and accuracy of this method. This method will become an effective tool in discovering the governing laws from available data sets and in understanding the mechanisms underlying complex random phenomena.