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Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach

arXiv 2105.04869 ↗First posted 11 May 2021cs.LG cs.CEcited by 36

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Abstract (arXiv)

Discovering dynamical models to describe underlying dynamical behavior is essential to draw decisive conclusions and engineering studies, e.g., optimizing a process. Experimental data availability notwithstanding has increased significantly, but interpretable and explainable models in science and engineering yet remain incomprehensible. In this work, we blend machine learning and dictionary-based learning with numerical analysis tools to discover governing differential equations from noisy and sparsely-sampled measurement data. We utilize the fact that given a dictionary containing huge candidate nonlinear functions, dynamical models can often be described by a few appropriately chosen candidates. As a result, we obtain interpretable and parsimonious models which are prone to generalize better beyond the sampling regime. Additionally, we integrate a numerical integration framework with dictionary learning that yields differential equations without requiring or approximating derivative information at any stage. Hence, it is utterly effective in corrupted and sparsely-sampled data. We discuss its extension to governing equations, containing rational nonlinearities that typically appear in biological networks. Moreover, we generalized the method to governing equations that are subject to parameter variations and externally controlled inputs. We demonstrate the efficiency of the method to discover a number of diverse differential equations using noisy measurements, including a model describing neural dynamics, chaotic Lorenz model, Michaelis-Menten Kinetics, and a parameterized Hopf normal form.

Key facts. Cited by 36 papers in the map (data through 7 Oct 2026). Builds on 6 of its 62 references that are in the map. First posted on arXiv 11 May 2021.

Builds on

6 of 62 references.

References · 6

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2016Densely Connected Convolutional Networkscited by 6,691
  3. 2018Neural Ordinary Differential Equationscited by 3,733
  4. 2018Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systemscited by 88
  5. 2018On the Convergence of the SINDy Algorithmcited by 49
  6. 2019A unified sparse optimization framework to learn parsimonious physics-informed models from datacited by 49

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2021: 3; 2022: 7; 2023: 6; 2024: 9; 2025: 8; 2026: 3.

36 papers.

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  1. 2022Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutionscited by 29
  2. 2021DySMHO: Data-Driven Discovery of Governing Equations for Dynamical Systems via Moving Horizon Optimizationcited by 19
  3. 2022Noise-aware Physics-informed Machine Learning for Robust PDE Discoverycited by 14
  4. 2024ODE Discovery for Longitudinal Heterogeneous Treatment Effects Inferencecited by 12
  5. 2022Operator inference with roll outs for learning reduced models from scarce and low-quality datacited by 11
  6. 2024Physics-Enhanced Machine Learning: a position paper for dynamical systems investigationscited by 11
  7. 2023Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inferencecited by 10
  8. 2024ADAM-SINDy: An Efficient Optimization Framework for Parameterized Nonlinear Dynamical System Identificationcited by 10
  9. 2022Generalized Quadratic Embeddings for Nonlinear Dynamics using Deep Learningcited by 9
  10. 2023Learning Closed-form Equations for Subgrid-scale Closures from High-fidelity Data: Promises and Challengescited by 9
  11. 2023AI-Lorenz: A physics-data-driven framework for black-box and gray-box identification of chaotic systems with symbolic regressioncited by 9
  12. 2021Learning Dynamics from Noisy Measurements using Deep Learning with a Runge-Kutta Constraintcited by 7
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. 2023Discussing the Spectrum of Physics-Enhanced Machine Learning; a Survey on Structural Mechanics Applicationscited by 6
  4. 2025SyMANTIC: An Efficient Symbolic Regression Method for Interpretable and Parsimonious Model Discovery in Science and Beyondcited by 6
  5. 2022Neural ODEs with Irregular and Noisy Datacited by 5
  6. 2024Sparse discovery of differential equations based on multi-fidelity Gaussian processcited by 5
  7. 2022SPADE4: Sparsity and Delay Embedding based Forecasting of Epidemicscited by 4
  8. 2024An invariance constrained deep learning network for PDE discoverycited by 3
  9. 2025Sparse identification of nonlinear dynamics with library optimization mechanism: Recursive long-term prediction perspectivecited by 2
  10. 2025Data-driven Discovery of Digital Twins in Biomedical Researchcited by 2
  11. 2026SINDy-KANs: Sparse identification of non-linear dynamics through Kolmogorov-Arnold networkscited by 2
  12. 2025Data driven feedback linearization of nonlinear control systems via Lie derivatives and stacked regression approachcited by 1
  13. 2025Data Denoising and Derivative Estimation for Data-Driven Modeling of Nonlinear Dynamical Systemscited by 1
  14. 2022Discovering Governing Equations by Machine Learning implemented with Invariancecited by 0
  15. 2023Inference of Continuous Linear Systems from Data with Guaranteed Stabilitycited by 0
  16. 2024Learning reduced-order Quadratic-Linear models in Process Engineering using Operator Inferencecited by 0
  17. 2024Stability-Certified Learning of Control Systems with Quadratic Nonlinearitiescited by 0
  18. 2024Fast Deep Predictive Coding Networks for Videos Feature Extraction without Labelscited by 0
  19. 2024Machine learning for cerebral blood vessels' malformationscited by 0
  20. 2025Impilict Runge-Kutta based sparse identification of governing equations in biologically motivated systemscited by 0
  21. 2025Discovering the dynamics of Sargassum rafts' centers of masscited by 0
  22. 2025From STLS to Projection-based Dictionary Selection in Sparse Regression for System Identificationcited by 0
  23. 2026Data Enrichment for Symbolic Regression Using Diffusion Modelscited by 0
  24. 2026Data-Driven Learning of Unknown Nonlinear Differential Equations Using Functional Analysiscited by 0