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SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics

arXiv 2004.02322 ↗First posted 5 Apr 2020cs.LG physics.comp-ph stat.MLcited by 125

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Abstract (arXiv)

Accurately modeling the nonlinear dynamics of a system from measurement data is a challenging yet vital topic. The sparse identification of nonlinear dynamics (SINDy) algorithm is one approach to discover dynamical systems models from data. Although extensions have been developed to identify implicit dynamics, or dynamics described by rational functions, these extensions are extremely sensitive to noise. In this work, we develop SINDy-PI (parallel, implicit), a robust variant of the SINDy algorithm to identify implicit dynamics and rational nonlinearities. The SINDy-PI framework includes multiple optimization algorithms and a principled approach to model selection. We demonstrate the ability of this algorithm to learn implicit ordinary and partial differential equations and conservation laws from limited and noisy data. In particular, we show that the proposed approach is several orders of magnitude more noise robust than previous approaches, and may be used to identify a class of complex ODE and PDE dynamics that were previously unattainable with SINDy, including for the double pendulum dynamics and the Belousov Zhabotinsky (BZ) reaction.

Key facts. Cited by 125 papers in the map (data through 7 Oct 2026). First posted on arXiv 5 Apr 2020.

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Citing papers per year: 2021: 10; 2022: 13; 2023: 24; 2024: 28; 2025: 21; 2026: 29.

125 papers.

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  1. 2021PySINDy: A comprehensive Python package for robust sparse system identificationcited by 88
  2. 2021Applications and Techniques for Fast Machine Learning in Sciencecited by 32
  3. 2022Learning Sparse Nonlinear Dynamics via Mixed-Integer Optimizationcited by 29
  4. 2023Latent Space Symmetry Discoverycited by 29
  5. 2023Uncertainty Quantification in Machine Learning for Engineering Design and Health Prognostics: A Tutorialcited by 27
  6. 2021Discovering Sparse Interpretable Dynamics from Partial Observationscited by 25
  7. 2022Bayesian autoencoders for data-driven discovery of coordinates, governing equations and fundamental constantscited by 24
  8. 2024Marrying Causal Representation Learning with Dynamical Systems for Sciencecited by 22
  9. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  10. 2023Controllable Neural Symbolic Regressioncited by 19
  11. 2021Physics-informed Spline Learning for Nonlinear Dynamics Discoverycited by 18
  12. 2022WeakIdent: Weak formulation for Identifying Differential Equations using Narrow-fit and Trimmingcited by 18
38 more
  1. 2022Discovery of partial differential equations from highly noisy and sparse data with physics-informed information criterioncited by 16
  2. 2023Extreme sparsification of physics-augmented neural networks for interpretable model discovery in mechanicscited by 16
  3. 2024Mechanistic Neural Networks for Scientific Machine Learningcited by 16
  4. 2021A toolkit for data-driven discovery of governing equations in high-noise regimescited by 15
  5. 2023Convergence of uncertainty estimates in Ensemble and Bayesian sparse model discoverycited by 15
  6. 2024Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy datacited by 15
  7. 2024LLM4ED: Large Language Models for Automatic Equation Discoverycited by 13
  8. 2025Generative Discovery of Partial Differential Equations by Learning from Math Handbookscited by 12
  9. 2022GSR: A Generalized Symbolic Regression Approachcited by 11
  10. 2023Interpretable learning of effective dynamics for multiscale systemscited by 11
  11. 2024Training Stiff Neural Ordinary Differential Equations with Implicit Single-Step Methodscited by 11
  12. 2023Pseudo-Hamiltonian neural networks for learning partial differential equationscited by 10
  13. 2024ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identificationcited by 10
  14. 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
  15. 2024Dynamical System Identification, Model Selection and Model Uncertainty Quantification by Bayesian Inferencecited by 9
  16. 2022Symbolic Recovery of Differential Equations: The Identifiability Problemcited by 8
  17. 2023Physics-constrained robust learning of open-form partial differential equations from limited and noisy datacited by 8
  18. 2021Inferring the Structure of Ordinary Differential Equationscited by 7
  19. 2022Reconstruction of observed mechanical motions with Artificial Intelligence toolscited by 7
  20. 2022D-CIPHER: Discovery of Closed-form Partial Differential Equationscited by 7
  21. 2023Bayesian polynomial neural networks and polynomial neural ordinary differential equationscited by 6
  22. 2023A Robust SINDy Approach by Combining Neural Networks and an Integral Formcited by 6
  23. 2023GINN-LP: A Growing Interpretable Neural Network for Discovering Multivariate Laurent Polynomial Equationscited by 6
  24. 2024BINDy -- Bayesian identification of nonlinear dynamics with reversible-jump Markov-chain Monte-Carlocited by 6
  25. 2025Discovering Symbolic Differential Equations with Symmetry Invariantscited by 6
  26. 2025When is a System Discoverable from Data? Discovery Requires Chaoscited by 6
  27. 2021Deep Learning of Conjugate Mappingscited by 5
  28. 2022textitFastSVD-ML-ROM: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applicationscited by 5
  29. 2024Scalable Mechanistic Neural Networks for Differential Equations and Machine Learningcited by 5
  30. 2024Training Stiff Neural Ordinary Differential Equations with Explicit Exponential Integration Methodscited by 5
  31. 2025Grammar-based Ordinary Differential Equation Discoverycited by 5
  32. 2025Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Modelscited by 5
  33. 2023Symbolic Regression on Sparse and Noisy Data with Gaussian Processescited by 4
  34. 2024Scalable Sparse Regression for Model Discovery: The Fast Lane to Insightcited by 4
  35. 2024Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspectivecited by 4
  36. 2024LES-SINDy: Laplace-Enhanced Sparse Identification of Nonlinear Dynamical Systemscited by 4
  37. 2024Recovering implicit physics model under real-world constraintscited by 4
  38. 2025Al-Khwarizmi: Discovering Physical Laws with Foundation Modelscited by 4

50 most cited of 125 All 125 (JSON)