mlsanityData to 14 Oct 2026

Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems

arXiv 2301.10321 ↗First posted 24 Jan 2023stat.ML cs.LGcited by 15

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Abstract (arXiv)

Regressing the vector field of a dynamical system from a finite number of observed states is a natural way to learn surrogate models for such systems. A simple and interpretable way to learn a dynamical system from data is to interpolate its vector-field with a data-adapted kernel which can be learned by using Kernel Flows. The method of Kernel Flows is a trainable machine learning method that learns the optimal parameters of a kernel based on the premise that a kernel is good if there is no significant loss in accuracy if half of the data is used. The objective function could be a short-term prediction or some other objective for other variants of Kernel Flows). However, this method is limited by the choice of the base kernel. In this paper, we introduce the method of Sparse Kernel Flows in order to learn the ``best'' kernel by starting from a large dictionary of kernels. It is based on sparsifying a kernel that is a linear combination of elemental kernels. We apply this approach to a library of 132 chaotic systems.

Key facts. Cited by 15 papers in the map (data through 14 Oct 2026). Builds on 18 of its 62 references that are in the map. First posted on arXiv 24 Jan 2023.

Builds on

18 of 62 references.

References · 18

  1. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 494
  2. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  3. 2021Chaos as an interpretable benchmark for forecasting and data-driven modellingcited by 88
  4. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  5. 2018Kernel Flows: from learning kernels from data into the abysscited by 49
  6. 2020Kernel-based approximation of the Koopman generator and Schr"odinger operatorcited by 31
  7. 2021Computational Graph Completioncited by 28
  8. 2020Learning dynamical systems from data: a simple cross-validation perspectivecited by 23
  9. 2021Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)cited by 22
  10. 2021Data-driven geophysical forecasting: Simple, low-cost, and accurate baselines with kernel methodscited by 15
  11. 2020Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theoremcited by 13
  12. 2019Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifoldscited by 12
6 more
  1. 2018A Note on Kernel Methods for Multiscale Systems with Critical Transitionscited by 11
  2. 2017Ensemble Kalman methods for high-dimensional hierarchical dynamic space-time modelscited by 10
  3. 2020Deep regularization and direct training of the inner layers of Neural Networks with Kernel Flowscited by 10
  4. 2019Data-driven prediction of a multi-scale Lorenz 96 chaotic system using deep learning methods: Reservoir computing, ANN, and RNN-LSTMcited by 9
  5. 2022Learning "best" kernels from data in Gaussian process regression. With application to aerodynamicscited by 9
  6. 2022Sparse dynamical system identification with simultaneous structural parameters and initial condition estimationcited by 7

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 6; 2025: 4; 2026: 3.

15 papers.

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  1. 2024Out-of-Domain Generalization in Dynamical Systems Reconstructioncited by 33
  2. 2024Zero-shot forecasting of chaotic systemscited by 28
  3. 2025Context parroting: A simple but tough-to-beat baseline for foundation models in scientific machine learningcited by 18
  4. 2024A scalable generative model for dynamical system reconstruction from neuroimaging datacited by 9
  5. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  6. 2024Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operatorcited by 5
  7. 2024Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approachcited by 4
  8. 2023Bridging Algorithmic Information Theory and Machine Learning: A New Approach to Kernel Learningcited by 3
  9. 2023Decoding Mean Field Games from Population and Environment Observations By Gaussian Processescited by 2
  10. 2025Learning Surrogate Potential Mean Field Games via Gaussian Processes: A Data-Driven Approach to Ill-Posed Inverse Problemscited by 2
  11. 2024Recurrent Deep Kernel Learning of Dynamical Systemscited by 1
  12. 2026Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximationcited by 1
3 more
  1. 2026Kernel Methods for Some Transport Equations with Application to Learning Kernels for the Approximation of Koopman Eigenfunctions: A Unified Approach via Variational Methods, Green's Functions and the Method of Characteristicscited by 1
  2. 2025Gaussian process policy iteration with additive Schwarz acceleration for forward and inverse HJB and mean field game problemscited by 0
  3. 2026Adaptive Kernel Selection for Kernelized Diffusion Mapscited by 0