mlsanityData to 7 Oct 2026

Extracting Sparse High-Dimensional Dynamics from Limited Data

arXiv 1707.08528 ↗First posted 26 Jul 2017math.OCcited by 50

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Abstract (arXiv)

Extracting governing equations from dynamic data is an essential task in model selection and parameter estimation. The form of the governing equation is rarely known a priori; however, based on the sparsity-of-effect principle one may assume that the number of candidate functions needed to represent the dynamics is very small. In this work, we leverage the sparse structure of the governing equations along with recent results from random sampling theory to develop methods for selecting dynamical systems from under-sampled data. In particular, we detail three sampling strategies that lead to the exact recovery of first-order dynamical systems when we are given fewer samples than unknowns. The first method makes no assumptions on the behavior of the data, and requires a certain number of random initial samples. The second method utilizes the structure of the governing equation to limit the number of random initializations needed. The third method leverages chaotic behavior in the data to construct a nearly deterministic sampling strategy. Using results from compressive sensing, we show that the strategies lead to exact recovery, which is stable to the sparse structure of the governing equations and robust to noise in the estimation of the velocity. Computational results validate each of the sampling strategies and highlight potential applications.

Key facts. Cited by 50 papers in the map (data through 7 Oct 2026). First posted on arXiv 26 Jul 2017.

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Citing papers per year: 2021: 14; 2022: 11; 2023: 10; 2024: 8; 2025: 5; 2026: 2.

50 papers.

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  10. 2023PROSE: Predicting Operators and Symbolic Expressions using Multimodal Transformerscited by 16
  11. 2024LeMON: Learning to Learn Multi-Operator Networkscited by 13
  12. 2023Reduced-Order Autodifferentiable Ensemble Kalman Filterscited by 12
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  10. 2021Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectoriescited by 5
  11. 2021Active operator inference for learning low-dimensional dynamical-system models from noisy datacited by 5
  12. 2022Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learningcited by 5
  13. 2024Statistical Mechanics of Dynamical System Identificationcited by 5
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  19. 2021Model discovery in the sparse sampling regimecited by 2
  20. 2021Adaptive Group Lasso Neural Network Models for Functions of Few Variables and Time-Dependent Datacited by 2
  21. 2023Data-Driven Model Selections of Second-Order Particle Dynamics via Integrating Gaussian Processes with Low-Dimensional Interacting Structurescited by 2
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  24. 2025Data-driven Discovery of Digital Twins in Biomedical Researchcited by 2
  25. 2021Uncovering Closed-form Governing Equations of Nonlinear Dynamics from Videoscited by 1
  26. 2021DeePN2: A deep learning-based non-Newtonian hydrodynamic modelcited by 1
  27. 2022Learning Fine Scale Dynamics from Coarse Observations via Inner Recurrencecited by 1
  28. 2026MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Datacited by 1
  29. 2026The Dynamics of Discoverability: How Trajectories and Priors Shape Equation Recoverycited by 1
  30. 2021Asymptotic Theory of ℓ1-Regularized PDE Identification from a Single Noisy Trajectorycited by 0
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  34. 2023Deep Learning based Spatially Dependent Acoustical Properties Recoverycited by 0
  35. 2023Improved identification accuracy in equation learning via comprehensive boldsymbolR2-elimination and Bayesian model selectioncited by 0
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  37. 2025A Sparse Bayesian Learning Algorithm for Estimation of Interaction Kernels in Motsch-Tadmor Modelcited by 0
  38. 2025An adaptive data sampling strategy for stabilizing dynamical systems via controller inferencecited by 0