# Benchmarking sparse system identification with low-dimensional chaos

arXiv: https://arxiv.org/abs/2302.10787 · HTML: https://mlsanity.com/paper/2302.10787/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 50 of its 157 references that are in the map.
- First posted on arXiv 4 Feb 2023.

## Builds on (most cited first, 30 of 50)

- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [Interpretable machine learning: definitions, methods, and applications](https://mlsanity.com/paper/1901.04592/) (2019)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Contemporary Symbolic Regression Methods and their Relative Performance](https://mlsanity.com/paper/2107.14351/) (2021)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Weak SINDy For Partial Differential Equations](https://mlsanity.com/paper/2007.02848/) (2020)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (2017)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [PySINDy: A comprehensive Python package for robust sparse system identification](https://mlsanity.com/paper/2111.08481/) (2021)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Sparse learning of stochastic dynamic equations](https://mlsanity.com/paper/1712.02432/) (2017)
- [On the difficulty of learning chaotic dynamics with RNNs](https://mlsanity.com/paper/2110.07238/) (2021)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [A unified sparse optimization framework to learn parsimonious physics-informed models from data](https://mlsanity.com/paper/1906.10612/) (2019)
- [Where are we now? A large benchmark study of recent symbolic regression methods](https://mlsanity.com/paper/1804.09331/) (2018)
- [Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data](https://mlsanity.com/paper/2009.08810/) (2020)
- [A Unified Framework for Sparse Relaxed Regularized Regression: SR3](https://mlsanity.com/paper/1807.05411/) (2018)
- [Tractable Dendritic RNNs for Reconstructing Nonlinear Dynamical Systems](https://mlsanity.com/paper/2207.02542/) (2022)
- [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)
- [Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions](https://mlsanity.com/paper/2211.06786/) (2022)
- [Sparse Regression: Scalable algorithms and empirical performance](https://mlsanity.com/paper/1902.06547/) (2019)
- [Data-Driven Modeling and Prediction of Non-Linearizable Dynamics via Spectral Submanifolds](https://mlsanity.com/paper/2201.04976/) (2022)

## Built on by (most cited first, 19 of 19)

- [Model scale versus domain knowledge in statistical forecasting of chaotic systems](https://mlsanity.com/paper/2303.08011/) (2023)
- [Weak-Form Latent Space Dynamics Identification](https://mlsanity.com/paper/2311.12880/) (2023)
- [Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy data](https://mlsanity.com/paper/2402.15357/) (2024)
- [Coarse-Graining Hamiltonian Systems Using WSINDy](https://mlsanity.com/paper/2310.05879/) (2023)
- [ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identification](https://mlsanity.com/paper/2410.16528/) (2024)
- [Detach-ROCKET: Sequential feature selection for time series classification with random convolutional kernels](https://mlsanity.com/paper/2309.14518/) (2023)
- [When is a System Discoverable from Data? Discovery Requires Chaos](https://mlsanity.com/paper/2511.08860/) (2025)
- [Data-driven local operator finding for reduced-order modelling of plasma systems: I. Concept and verifications](https://mlsanity.com/paper/2403.01523/) (2024)
- [Statistical Mechanics of Dynamical System Identification](https://mlsanity.com/paper/2403.01723/) (2024)
- [Symbolic--KAN: Kolmogorov-Arnold Networks with Discrete Symbolic Structure for Interpretable Learning](https://mlsanity.com/paper/2603.23854/) (2026)
- [Sparse Identification of Nonlinear Dynamics with Conformal Prediction](https://mlsanity.com/paper/2507.11739/) (2025)
- [Data-driven discovery of governing differential equations across physical systems](https://mlsanity.com/paper/2606.09638/) (2026)
- [Sparse identification of nonlinear dynamics with library optimization mechanism: Recursive long-term prediction perspective](https://mlsanity.com/paper/2507.18220/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recovery](https://mlsanity.com/paper/2607.18490/) (2026)
- [System Identification with Copula Entropy](https://mlsanity.com/paper/2304.12922/) (2023)
- [Impilict Runge-Kutta based sparse identification of governing equations in biologically motivated systems](https://mlsanity.com/paper/2502.20319/) (2025)
- [A tensor network approach for chaotic time series prediction](https://mlsanity.com/paper/2505.17740/) (2025)
- [SINDy on slow manifolds](https://mlsanity.com/paper/2507.00747/) (2025)

## Abstract (arXiv)

Sparse system identification is the data-driven process of obtaining parsimonious differential equations that describe the evolution of a dynamical system, balancing model complexity and accuracy. There has been rapid innovation in system identification across scientific domains, but there remains a gap in the literature for large-scale methodological comparisons that are evaluated on a variety of dynamical systems. In this work, we systematically benchmark sparse regression variants by utilizing the dysts standardized database of chaotic systems. In particular, we demonstrate how this open-source tool can be used to quantitatively compare different methods of system identification. To illustrate how this benchmark can be utilized, we perform a large comparison of four algorithms for solving the sparse identification of nonlinear dynamics (SINDy) optimization problem, finding strong performance of the original algorithm and a recent mixed-integer discrete algorithm. In all cases, we used ensembling to improve the noise robustness of SINDy and provide statistical comparisons. In addition, we show very compelling evidence that the weak SINDy formulation provides significant improvements over the traditional method, even on clean data. Lastly, we investigate how Pareto-optimal models generated from SINDy algorithms depend on the properties of the equations, finding that the performance shows no significant dependence on a set of dynamical properties that quantify the amount of chaos, scale separation, degree of nonlinearity, and the syntactic complexity.