Extracting stochastic dynamical systems with $\alpha$-stable L\'evy noise from data

TL;DR

This paper introduces a data-driven method to identify stochastic dynamical systems influenced by non-Gaussian $b5$-stable Le9vy noise from short data sequences, combining statistical estimation and neural network techniques.

Contribution

The authors develop a novel approach that estimates Le9vy noise parameters and drift coefficients from limited data, advancing the analysis of complex systems with non-Gaussian fluctuations.

Findings

Accurately estimates Le9vy jump measures and noise intensity from data.

Effectively reconstructs drift coefficients using nonlocal Kramers-Moyal formulas and normalizing flows.

Demonstrates high accuracy and robustness in one- and two-dimensional examples.

Abstract

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) L\'evy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with $\alpha$-stable L\'evy noise from short burst data based on the properties of $\alpha$-stable distributions. More specifically, we first estimate the L\'evy 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 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.