mlsanityData to 7 Oct 2026

PySINDy: A comprehensive Python package for robust sparse system identification

arXiv 2111.08481 ↗First posted 12 Nov 2021eess.SY cs.LG cs.SYcited by 88

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Abstract (arXiv)

Automated data-driven modeling, the process of directly discovering the governing equations of a system from data, is increasingly being used across the scientific community. PySINDy is a Python package that provides tools for applying the sparse identification of nonlinear dynamics (SINDy) approach to data-driven model discovery. In this major update to PySINDy, we implement several advanced features that enable the discovery of more general differential equations from noisy and limited data. The library of candidate terms is extended for the identification of actuated systems, partial differential equations (PDEs), and implicit differential equations. Robust formulations, including the integral form of SINDy and ensembling techniques, are also implemented to improve performance for real-world data. Finally, we provide a range of new optimization algorithms, including several sparse regression techniques and algorithms to enforce and promote inequality constraints and stability. Together, these updates enable entirely new SINDy model discovery capabilities that have not been reported in the literature, such as constrained PDE identification and ensembling with different sparse regression optimizers.

Key facts. Cited by 88 papers in the map (data through 7 Oct 2026). Builds on 12 of its 39 references that are in the map. First posted on arXiv 12 Nov 2021.

Builds on

12 of 39 references.

References · 12

  1. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  2. 2018Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networkscited by 155
  3. 2020Weak SINDy For Partial Differential Equationscited by 154
  4. 2017Machine Learning of Linear Differential Equations using Gaussian Processescited by 139
  5. 2017Sparse identification of nonlinear dynamics for model predictive control in the low-data limitcited by 139
  6. 2020SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamicscited by 125
  7. 2019Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systemscited by 96
  8. 2017Sparse learning of stochastic dynamic equationscited by 76
  9. 2019A unified sparse optimization framework to learn parsimonious physics-informed models from datacited by 49
  10. 2021SINDy with Control: A Tutorialcited by 15
  11. 2021A toolkit for data-driven discovery of governing equations in high-noise regimescited by 15
  12. 2019Stability selection enables robust learning of partial differential equations from limited noisy datacited by 7

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 9; 2023: 21; 2024: 17; 2025: 20; 2026: 21.

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  1. 2023Machine learning with data assimilation and uncertainty quantification for dynamical systems: a reviewcited by 74
  2. 2022Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoderscited by 58
  3. 2023Generalized Teacher Forcing for Learning Chaotic Dynamicscited by 55
  4. 2023Interpretable Machine Learning for Science with PySR and SymbolicRegression.jlcited by 36
  5. 2022SIMPLE: A Gradient Estimator for k-Subset Samplingcited by 34
  6. 2022A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledgecited by 31
  7. 2022Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimizationcited by 29
  8. 2022Learning continuous models for continuous physicscited by 28
  9. 2022Interpretable Polynomial Neural Ordinary Differential Equationscited by 24
  10. 2023Direct Estimation of Parameters in ODE Models Using WENDy: Weak-form Estimation of Nonlinear Dynamicscited by 20
  11. 2023Machine Learning for Partial Differential Equationscited by 20
  12. 2023Weak-Form Latent Space Dynamics Identificationcited by 20
38 more
  1. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  2. 2023Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discoverycited by 15
  3. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  4. 2026Think like a Scientist: Physics-guided LLM Agent for Equation Discoverycited by 13
  5. 2023PyKoopman: A Python Package for Data-Driven Approximation of the Koopman Operatorcited by 10
  6. 2024Shallow Recurrent Decoder for Reduced Order Modeling of Plasma Dynamicscited by 8
  7. 2022D-CIPHER: Discovery of Closed-form Partial Differential Equationscited by 7
  8. 2023OKRidge: Scalable Optimal k-Sparse Ridge Regressioncited by 7
  9. 2023Detach-ROCKET: Sequential feature selection for time series classification with random convolutional kernelscited by 7
  10. 2023HyperSINDy: Deep Generative Modeling of Nonlinear Stochastic Governing Equationscited by 7
  11. 2024Constrained Exploration via Reflected Replica Exchange Stochastic Gradient Langevin Dynamicscited by 7
  12. 2025Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Modelscited by 7
  13. 2023Adaptive Uncertainty-Guided Model Selection for Data-Driven PDE Discoverycited by 6
  14. 2024SymbolNet: Neural Symbolic Regression with Adaptive Dynamic Pruning for Compressioncited by 6
  15. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  16. 2023ConCerNet: A Contrastive Learning Based Framework for Automated Conservation Law Discovery and Trustworthy Dynamical System Predictioncited by 5
  17. 2023Learning noise-induced transitions by multi-scaling reservoir computingcited by 5
  18. 2023Symbolic Regression as Feature Engineering Method for Machine and Deep Learning Regression Taskscited by 5
  19. 2024Sparse discovery of differential equations based on multi-fidelity Gaussian processcited by 5
  20. 2022Model-Based Reinforcement Learning with SINDycited by 4
  21. 2024LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systemscited by 4
  22. 2025Al-Khwarizmi: Discovering Physical Laws with Foundation Modelscited by 4
  23. 2024Closed-Form Interpretation of Neural Network Classifiers with Symbolic Gradientscited by 3
  24. 2024Unit-Aware Genetic Programming for the Development of Empirical Equationscited by 3
  25. 2024Constrained or Unconstrained? Neural-Network-Based Equation Discovery from Datacited by 3
  26. 2025Information theory and discriminative sampling for model discoverycited by 3
  27. 2026Data-driven discovery of governing differential equations across physical systemscited by 3
  28. 2023Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculuscited by 2
  29. 2024Stabilizing black-box model selection with the inflated argmaxcited by 2
  30. 2024Interpretable and Efficient Data-driven Discovery and Control of Distributed Systemscited by 2
  31. 2025No Equations Needed: Learning System Dynamics Without Relying on Closed-Form ODEscited by 2
  32. 2025Discovering equations from data: symbolic regression in dynamical systemscited by 2
  33. 2025Data-driven Discovery of Digital Twins in Biomedical Researchcited by 2
  34. 2025Beyond empirical models: Discovering new constitutive laws in solids with graph-based equation discoverycited by 2
  35. 2026SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networkscited by 2
  36. 2026Data-driven sparse identification of governing PDEs via knockoff filters and multi-criteria trade-offscited by 2
  37. 2024UniFIDES: Universal Fractional Integro-Differential Equation Solverscited by 1
  38. 2024Solving the 2D Advection-Diffusion Equation using Fixed-Depth Symbolic Regression and Symbolic Differentiation without Expression Treescited by 1

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