# Weak Collocation Regression for Inferring Stochastic Dynamics with L'evy Noise

arXiv: https://arxiv.org/abs/2403.08292 · HTML: https://mlsanity.com/paper/2403.08292/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 48 references that are in the map.
- First posted on arXiv 13 Mar 2024.

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

- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [SDE-Net: Equipping Deep Neural Networks with Uncertainty Estimates](https://mlsanity.com/paper/2008.10546/) (2020)
- [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)
- [OnsagerNet: Learning Stable and Interpretable Dynamics using a Generalized Onsager Principle](https://mlsanity.com/paper/2009.02327/) (2020)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Revealing hidden dynamics from time-series data by ODENet](https://mlsanity.com/paper/2005.04849/) (2020)
- [A Data-Driven Approach for Discovering Stochastic Dynamical Systems with Non-Gaussian Levy Noise](https://mlsanity.com/paper/2005.03769/) (2020)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [Non-parametric Estimation of Stochastic Differential Equations with Sparse Gaussian Processes](https://mlsanity.com/paper/1704.04375/) (2017)
- [Learning Stochastic Behaviour from Aggregate Data](https://mlsanity.com/paper/2002.03513/) (2020)
- [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)
- [Generative Ensemble Regression: Learning Particle Dynamics from Observations of Ensembles with Physics-Informed Deep Generative Models](https://mlsanity.com/paper/2008.01915/) (2020)
- [Weak Collocation Regression method: fast reveal hidden stochastic dynamics from high-dimensional aggregate data](https://mlsanity.com/paper/2209.02628/) (2022)
- [Drift Identification for L'evy alpha-Stable Stochastic Systems](https://mlsanity.com/paper/2212.03317/) (2022)

## Built on by (most cited first, 1 of 1)

- [A Robust Model-Based Approach for Continuous-Time Policy Evaluation with Unknown L'evy Process Dynamics](https://mlsanity.com/paper/2504.01482/) (2025)

## Abstract (arXiv)

With the rapid increase of observational, experimental and simulated data for stochastic systems, tremendous efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the broad applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extracting stochastic dynamics with L'evy noise are relatively few. In this work, we propose a Weak Collocation Regression (WCR) to explicitly reveal unknown stochastic dynamical systems, i.e., the Stochastic Differential Equation (SDE) with both α-stable L'evy noise and Gaussian noise, from discrete aggregate data. This method utilizes the evolution equation of the probability distribution function, i.e., the Fokker-Planck (FP) equation. With the weak form of the FP equation, the WCR constructs a linear system of unknown parameters where all integrals are evaluated by Monte Carlo method with the observations. Then, the unknown parameters are obtained by a sparse linear regression. For a SDE with L'evy noise, the corresponding FP equation is a partial integro-differential equation (PIDE), which contains nonlocal terms, and is difficult to deal with. The weak form can avoid complicated multiple integrals. Our approach can simultaneously distinguish mixed noise types, even in multi-dimensional problems. Numerical experiments demonstrate that our method is accurate and computationally efficient.