# An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noise

arXiv: https://arxiv.org/abs/2201.13114 · HTML: https://mlsanity.com/paper/2201.13114/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 65 references that are in the map.
- First posted on arXiv 31 Jan 2022.

## Builds on (most cited first, 16 of 16)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Neural ODE Processes](https://mlsanity.com/paper/2103.12413/) (2021)
- [Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networks](https://mlsanity.com/paper/2008.10653/) (2020)
- [Black-box Variational Inference for Stochastic Differential Equations](https://mlsanity.com/paper/1802.03335/) (2018)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes](https://mlsanity.com/paper/1704.04375/) (2017)
- [Learning stochastic dynamical systems with neural networks mimicking the Euler-Maruyama scheme](https://mlsanity.com/paper/2105.08449/) (2021)
- [Extracting stochastic dynamical systems with α-stable L'evy noise from data](https://mlsanity.com/paper/2109.14881/) (2021)
- [Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems](https://mlsanity.com/paper/2107.10127/) (2021)
- [Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas](https://mlsanity.com/paper/2108.12570/) (2021)
- [Recognizing Car Fluents from Video](https://mlsanity.com/paper/1603.08067/) (2016)

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

- [Early warning indicators via latent stochastic dynamical systems](https://mlsanity.com/paper/2309.03842/) (2023)
- [Tasks Makyth Models: Machine Learning Assisted Surrogates for Tipping Points](https://mlsanity.com/paper/2309.14334/) (2023)
- [Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systems](https://mlsanity.com/paper/2305.00669/) (2023)
- [Drift Identification for L'evy alpha-Stable Stochastic Systems](https://mlsanity.com/paper/2212.03317/) (2022)
- [Tipping Points of Evolving Epidemiological Networks: Machine Learning-Assisted, Data-Driven Effective Modeling](https://mlsanity.com/paper/2311.00797/) (2023)
- [Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noise](https://mlsanity.com/paper/2403.08292/) (2024)
- [Inference for stochastic differential equations driven by weighted sub-fractional Brownian motion using neural networks and the Euler approximation](https://mlsanity.com/paper/2610.00793/) (2026)

## Abstract (arXiv)

Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained a lot of attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, lots of log-likelihood based algorithms that work well for Gaussian cases cannot be directly extended to non-Gaussian scenarios which could have high error and low convergence issues. In this work, we overcome some of these challenges and identify stochastic dynamical systems driven by α-stable L'evy noise from only random pairwise data. Our innovations include: (1) designing a deep learning approach to learn both drift and diffusion coefficients for L'evy induced noise with α across all values, (2) learning complex multiplicative noise without restrictions on small noise intensity, (3) proposing an end-to-end complete framework for stochastic systems identification under a general input data assumption, that is, α-stable random variable. Finally, numerical experiments and comparisons with the non-local Kramers-Moyal formulas with moment generating function confirm the effectiveness of our method.