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Automatic Differentiation to Simultaneously Identify Nonlinear Dynamics and Extract Noise Probability Distributions from Data

arXiv 2009.08810 ↗First posted 12 Sep 2020eess.SP cs.LG math.DScited by 36

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Abstract (arXiv)

The sparse identification of nonlinear dynamics (SINDy) is a regression framework for the discovery of parsimonious dynamic models and governing equations from time-series data. As with all system identification methods, noisy measurements compromise the accuracy and robustness of the model discovery procedure. In this work, we develop a variant of the SINDy algorithm that integrates automatic differentiation and recent time-stepping constrained motivated by Rudy et al. for simultaneously (i) denoising the data, (ii) learning and parametrizing the noise probability distribution, and (iii) identifying the underlying parsimonious dynamical system responsible for generating the time-series data. Thus within an integrated optimization framework, noise can be separated from signal, resulting in an architecture that is approximately twice as robust to noise as state-of-the-art methods, handling as much as 40% noise on a given time-series signal and explicitly parametrizing the noise probability distribution. We demonstrate this approach on several numerical examples, from Lotka-Volterra models to the spatio-temporal Lorenz 96 model. Further, we show the method can identify a diversity of probability distributions including Gaussian, uniform, Gamma, and Rayleigh.

Key facts. Cited by 36 papers in the map (data through 7 Oct 2026). First posted on arXiv 12 Sep 2020.

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 5; 2022: 6; 2023: 7; 2024: 7; 2025: 5; 2026: 6.

36 papers.

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  1. 2021A Framework for Machine Learning of Model Error in Dynamical Systemscited by 37
  2. 2023Discovering Causal Relations and Equations from Datacited by 33
  3. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  4. 2021Discovering Sparse Interpretable Dynamics from Partial Observationscited by 25
  5. 2021DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimizationcited by 19
  6. 2023Benchmarking sparse system identification with low-dimensional chaoscited by 19
  7. 2024Almost-Linear RNNs Yield Highly Interpretable Symbolic Codes in Dynamical Systems Reconstructioncited by 18
  8. 2022Discrepancy Modeling Framework: Learning missing physics, modeling systematic residuals, and disambiguating between deterministic and random effectscited by 15
  9. 2024Rapid Bayesian identification of sparse nonlinear dynamics from scarce and noisy datacited by 15
  10. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  11. 2022Asymptotic consistency of the WSINDy algorithm in the limit of continuum datacited by 14
  12. 2024On latent dynamics learning in nonlinear reduced order modelingcited by 14
24 more
  1. 2021Discovery of interpretable structural model errors by combining Bayesian sparse regression and data assimilation: A chaotic Kuramoto-Sivashinsky test casecited by 6
  2. 2023A Robust SINDy Approach by Combining Neural Networks and an Integral Formcited by 6
  3. 2022A Priori Denoising Strategies for Sparse Identification of Nonlinear Dynamical Systems: A Comparative Studycited by 5
  4. 2023Learning noise-induced transitions by multi-scaling reservoir computingcited by 5
  5. 2024Sparse discovery of differential equations based on multi-fidelity Gaussian processcited by 5
  6. 2023Symbolic Regression on Sparse and Noisy Data with Gaussian Processescited by 4
  7. 2024Machine Learning Applications to Computational Plasma Physics and Reduced-Order Plasma Modeling: A Perspectivecited by 4
  8. 2023Multiphysics discovery with moving boundaries using Ensemble SINDy and Peridynamic Differential Operatorcited by 3
  9. 2025Sparse Identification of Nonlinear Dynamics with Conformal Predictioncited by 3
  10. 2022Investigating Sindy As a Tool For Causal Discovery In Time Series Signalscited by 2
  11. 2025Data-driven Discovery of Digital Twins in Biomedical Researchcited by 2
  12. 2025Interpretable neural network system identification method for two families of second-order systems based on characteristic curvescited by 2
  13. 2021Uncovering Closed-form Governing Equations of Nonlinear Dynamics from Videoscited by 1
  14. 2022Digital Twin Data Modelling by Randomized Orthogonal Decomposition and Deep Learningcited by 1
  15. 2024Data-driven discovery of mechanical models directly from MRI spectral datacited by 1
  16. 2025Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systemscited by 1
  17. 2026Fourier Weak SINDy: Spectral Test Function Selection for Robust Model Identificationcited by 1
  18. 2026LLM-PDESR: Robust PDE Discovery via Subdomain Weighted Residuals and LLM-Guided Symbolic Hypothesis Generationcited by 1
  19. 2023Improved identification accuracy in equation learning via comprehensive boldsymbolR2-elimination and Bayesian model selectioncited by 0
  20. 2025A joint optimization approach to identifying sparse dynamics using least squares kernel collocationcited by 0
  21. 2026Towards a data-scale independent regulariser for robust sparse identification of non-linear dynamicscited by 0
  22. 2026Balance-Guided Sparse Identification of Multiscale Nonlinear PDEs with Small-coefficient Termscited by 0
  23. 2026Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysiscited by 0
  24. 2026Tensor-Train Weak SINDy: Identifying High-Dimensional Nonlinear Dynamicscited by 0