Extracting stochastic dynamical systems with $α$-stable Lévy noise from data
This work addresses the challenge of identifying governing laws in complex systems with non-Gaussian fluctuations, which is incremental as it builds on existing data-driven approaches but focuses on Lévy noise.
The authors tackled the problem of extracting stochastic dynamical systems with α-stable Lévy noise from short burst data, proposing a data-driven method that estimates the Lévy jump measure and noise intensity, and approximates the drift coefficient using nonlocal Kramers-Moyal formulas with normalizing flows, demonstrating accuracy and effectiveness in numerical experiments on one- and two-dimensional examples.
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) Lévy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with $α$-stable Lévy noise from short burst data based on the properties of $α$-stable distributions. More specifically, we first estimate the Lévy 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.