# Sparse learning of stochastic dynamic equations

arXiv: https://arxiv.org/abs/1712.02432 · HTML: https://mlsanity.com/paper/1712.02432/

## Key facts

- Cited by 76 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2017.

## Built on by (most cited first, 30 of 76)

- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Learning Mean-Field Equations from Particle Data Using WSINDy](https://mlsanity.com/paper/2110.07756/) (2021)
- [Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimization](https://mlsanity.com/paper/2206.00176/) (2022)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [PDE-READ: Human-readable Partial Differential Equation Discovery using Deep Learning](https://mlsanity.com/paper/2111.00998/) (2021)
- [Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constants](https://mlsanity.com/paper/2211.10575/) (2022)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterion](https://mlsanity.com/paper/2208.03322/) (2022)
- [A toolkit for data-driven discovery of governing equations in high-noise regimes](https://mlsanity.com/paper/2111.04870/) (2021)
- [Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discovery](https://mlsanity.com/paper/2301.12649/) (2023)
- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [Learning About Structural Errors in Models of Complex Dynamical Systems](https://mlsanity.com/paper/2401.00035/) (2023)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [Accurate Machine Learned Quantum-Mechanical Force Fields for Biomolecular Simulations](https://mlsanity.com/paper/2205.08306/) (2022)
- [Extracting stochastic dynamical systems with α-stable L'evy noise from data](https://mlsanity.com/paper/2109.14881/) (2021)
- [A Causality-Based Learning Approach for Discovering the Underlying Dynamics of Complex Systems from Partial Observations with Stochastic Parameterization](https://mlsanity.com/paper/2208.09104/) (2022)
- [Detach-ROCKET: Sequential feature selection for time series classification with random convolutional kernels](https://mlsanity.com/paper/2309.14518/) (2023)
- [HyperSINDy: Deep Generative Modeling of Nonlinear Stochastic Governing Equations](https://mlsanity.com/paper/2310.04832/) (2023)
- [Extracting Governing Laws from Sample Path Data of Non-Gaussian Stochastic Dynamical Systems](https://mlsanity.com/paper/2107.10127/) (2021)
- [D-CIPHER: Discovery of Closed-form Partial Differential Equations](https://mlsanity.com/paper/2206.10586/) (2022)
- [OKRidge: Scalable Optimal k-Sparse Ridge Regression](https://mlsanity.com/paper/2304.06686/) (2023)
- [In-Context Learning of Stochastic Differential Equations with Foundation Inference Models](https://mlsanity.com/paper/2502.19049/) (2025)
- [Discovering stochastic partial differential equations from limited data using variational Bayes inference](https://mlsanity.com/paper/2306.15873/) (2023)
- [Equivariance and partial observations in Koopman operator theory for partial differential equations](https://mlsanity.com/paper/2307.15325/) (2023)
- [A Robust SINDy Approach by Combining Neural Networks and an Integral Form](https://mlsanity.com/paper/2309.07193/) (2023)
- [Data assimilation and discrepancy modeling with shallow recurrent decoders](https://mlsanity.com/paper/2512.01170/) (2025)

## Abstract (arXiv)

With the rapid increase of available data for complex systems, there is great interest in the extraction of physically relevant information from massive datasets. Recently, a framework called Sparse Identification of Nonlinear Dynamics (SINDy) has been introduced to identify the governing equations of dynamical systems from simulation data. In this study, we extend SINDy to stochastic dynamical systems, which are frequently used to model biophysical processes. We prove the asymptotic correctness of stochastics SINDy in the infinite data limit, both in the original and projected variables. We discuss algorithms to solve the sparse regression problem arising from the practical implementation of SINDy, and show that cross validation is an essential tool to determine the right level of sparsity. We demonstrate the proposed methodology on two test systems, namely, the diffusion in a one-dimensional potential, and the projected dynamics of a two-dimensional diffusion process.