mlsanityData to 14 Oct 2026

Extracting Stochastic Governing Laws by Nonlocal Kramers-Moyal Formulas

arXiv 2108.12570 ↗First posted 28 Aug 2021math.DS math.PR stat.MLcited by 6

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Abstract (arXiv)

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian L'evy noise are relatively few so far. In this work, we propose a data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) L'evy motion, from short bursts of simulation data. Specifically, we use the normalizing flows technology to estimate the transition probability density function (solution of nonlocal Fokker-Planck equation) from data, and then substitute it into the recently proposed nonlocal Kramers-Moyal formulas to approximate L'evy jump measure, drift coefficient and diffusion coefficient. We demonstrate that this approach can learn the stochastic differential equation with L'evy motion. We present examples with one- and two-dimensional, decoupled and coupled systems to illustrate our method. This approach will become an effective tool for discovering stochastic governing laws and understanding complex dynamical behaviors.

Key facts. Cited by 6 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 28 Aug 2021.

Builds on

16 of 65 references.

References · 16

  1. 2018Neural Ordinary Differential Equationscited by 3,779
  2. 2015Variational Inference with Normalizing Flowscited by 2,170
  3. 2016Density estimation using Real NVPcited by 2,042
  4. 2019Normalizing Flows for Probabilistic Modeling and Inferencecited by 1,323
  5. 2017Masked Autoregressive Flow for Density Estimationcited by 684
  6. 2016Improving Variational Inference with Inverse Autoregressive Flowcited by 640
  7. 2020Scalable Gradients for Stochastic Differential Equationscited by 271
  8. 2019Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limitcited by 176
  9. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  10. 2020Solving Inverse Stochastic Problems from Discrete Particle Observations Using the Fokker-Planck Equation and Physics-informed Neural Networkscited by 52
  11. 2019Coarse-scale PDEs from fine-scale observations via machine learningcited by 35
  12. 2021Learning the temporal evolution of multivariate densities via normalizing flowscited by 18
4 more
  1. 2020A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noisecited by 15
  2. 2017Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processescited by 12
  3. 2019Temporal Normalizing Flowscited by 8
  4. 2020Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-Informed Deep Generative Modelscited by 6

Built on by

Cited by year

Citing papers per year: 2021: 1; 2022: 3; 2023: 1; 2024: 1.

6 papers.

  1. 2021Extracting stochastic dynamical systems with α-stable L'evy noise from datacited by 8
  2. 2022An end-to-end deep learning approach for extracting stochastic dynamical systems with α-stable L'evy noisecited by 7
  3. 2022Learning Effective SDEs from Brownian Dynamics Simulations of Colloidal Particlescited by 5
  4. 2023Reservoir Computing with Error Correction: Long-term Behaviors of Stochastic Dynamical Systemscited by 2
  5. 2022Drift Identification for L'evy alpha-Stable Stochastic Systemscited by 1
  6. 2024An evolutionary approach for discovering non-Gaussian stochastic dynamical systems based on nonlocal Kramers-Moyal formulascited by 1