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

arXiv: https://arxiv.org/abs/2107.10127 · HTML: https://mlsanity.com/paper/2107.10127/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 66 references that are in the map.
- First posted on arXiv 21 Jul 2021.

## Builds on (most cited first, 4 of 4)

- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [Extracting Non-Gaussian Governing Laws from Data on Mean Exit Time](https://mlsanity.com/paper/2006.14974/) (2020)

## Built on by (most cited first, 7 of 7)

- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Extracting stochastic dynamical systems with α-stable L'evy noise from data](https://mlsanity.com/paper/2109.14881/) (2021)
- [An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise](https://mlsanity.com/paper/2201.13114/) (2022)
- [Early warning indicators via latent stochastic dynamical systems](https://mlsanity.com/paper/2309.03842/) (2023)
- [Drift Identification for L'evy alpha-Stable Stochastic Systems](https://mlsanity.com/paper/2212.03317/) (2022)
- [An evolutionary approach for discovering non-Gaussian stochastic dynamical systems based on nonlocal Kramers-Moyal formulas](https://mlsanity.com/paper/2409.19534/) (2024)
- [Nonlocal Kramers-Moyal formulas and data-driven discovery of stochastic dynamical systems with multiplicative L'evy noise](https://mlsanity.com/paper/2601.19223/) (2026)

## Abstract (arXiv)

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.