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Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces

arXiv 2205.14027 ↗First posted 27 May 2022cs.LG math.DScited by 66

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Abstract (arXiv)

We study a class of dynamical systems modelled as Markov chains that admit an invariant distribution via the corresponding transfer, or Koopman, operator. While data-driven algorithms to reconstruct such operators are well known, their relationship with statistical learning is largely unexplored. We formalize a framework to learn the Koopman operator from finite data trajectories of the dynamical system. We consider the restriction of this operator to a reproducing kernel Hilbert space and introduce a notion of risk, from which different estimators naturally arise. We link the risk with the estimation of the spectral decomposition of the Koopman operator. These observations motivate a reduced-rank operator regression (RRR) estimator. We derive learning bounds for the proposed estimator, holding both in i.i.d. and non i.i.d. settings, the latter in terms of mixing coefficients. Our results suggest RRR might be beneficial over other widely used estimators as confirmed in numerical experiments both for forecasting and mode decomposition.

Key facts. Cited by 66 papers in the map (data through 7 Oct 2026). Builds on 12 of its 81 references that are in the map. First posted on arXiv 27 May 2022.

Builds on

12 of 81 references.

References · 12

  1. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  2. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 360
  3. 2021Modern Koopman Theory for Dynamical Systemscited by 269
  4. 2017VAMPnets: Deep learning of molecular kineticscited by 135
  5. 2020Kernel methods through the roof: handling billions of points efficientlycited by 88
  6. 2017Time-lagged autoencoders: Deep learning of slow collective variables for molecular kineticscited by 64
  7. 2017Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spacescited by 48
  8. 2021Learning Stable Koopman Embeddingscited by 18
  9. 2021Learning to Forecast Dynamical Systems from Streaming Datacited by 17
  10. 2018Singular Value Decomposition of Operators on Reproducing Kernel Hilbert Spacescited by 14
  11. 2019Leveraging Low-Rank Relations Between Surrogate Tasks in Structured Predictioncited by 5
  12. 2020Reduced Rank Multivariate Kernel Ridge Regressioncited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2022: 4; 2023: 14; 2024: 16; 2025: 16; 2026: 16.

66 papers.

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  1. 2023Resurrecting Recurrent Neural Networks for Long Sequencescited by 397
  2. 2024Operator Learning: Algorithms and Analysiscited by 84
  3. 2023Sharp Spectral Rates for Koopman Operator Learningcited by 35
  4. 2023Discovering Causal Relations and Equations from Datacited by 33
  5. 2022Learning linear operators: Infinite-dimensional regression as a well-behaved non-compact inverse problemcited by 27
  6. 2023Koopman Kernel Regressioncited by 22
  7. 2023Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithmcited by 22
  8. 2024Operator Learning Using Random Features: A Tool for Scientific Computingcited by 21
  9. 2023Learning invariant representations of time-homogeneous stochastic dynamical systemscited by 17
  10. 2023Estimating Koopman operators with sketching to provably learn large scale dynamical systemscited by 15
  11. 2022Spectral Representation Learning for Conditional Moment Modelscited by 12
  12. 2024Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithmscited by 12
38 more
  1. 2023Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systemscited by 11
  2. 2023Consistent Long-Term Forecasting of Ergodic Dynamical Systemscited by 10
  3. 2023Neural Koopman prior for data assimilationcited by 9
  4. 2024Learning the Infinitesimal Generator of Stochastic Diffusion Processescited by 9
  5. 2024Nonparametric Control Koopman Operatorscited by 7
  6. 2024Operator World Models for Reinforcement Learningcited by 7
  7. 2024Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroupscited by 7
  8. 2024Nonparametric Instrumental Regression via Kernel Methods is Minimax Optimalcited by 7
  9. 2025Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systemscited by 7
  10. 2024From Biased to Unbiased Dynamics: An Infinitesimal Generator Approachcited by 6
  11. 2025Koopman-Equivariant Gaussian Processescited by 6
  12. 2025Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systemscited by 5
  13. 2025Universal Learning of Nonlinear Dynamicscited by 5
  14. 2023A randomized algorithm to solve reduced rank operator regressioncited by 4
  15. 2024SafEDMD: A Koopman-based data-driven controller design framework for nonlinear dynamical systemscited by 4
  16. 2023Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modellingcited by 3
  17. 2024Long-Context Linear System Identificationcited by 3
  18. 2025Information Shapes Koopman Representationcited by 3
  19. 2024Linear quadratic control of nonlinear systems with Koopman operator learning and the Nystr"om methodcited by 2
  20. 2024Data-Driven Stochastic Optimal Control in Reproducing Kernel Hilbert Spacescited by 2
  21. 2025Deep Koopman operator framework for causal discovery in nonlinear dynamical systemscited by 2
  22. 2025A Spectral-Grassmann Wasserstein metric for operator representations of dynamical systemscited by 2
  23. 2025AMORE: Adaptive Multi-Output Operator Network for Stiff Chemical Kineticscited by 2
  24. 2025Sequence Modeling with Spectral Mean Flowscited by 2
  25. 2022Ensemble forecasts in reproducing kernel Hilbert space familycited by 1
  26. 2024Nonparametric Sparse Online Learning of the Koopman Operatorcited by 1
  27. 2024Kernel-Based Optimal Control: An Infinitesimal Generator Approachcited by 1
  28. 2025A Kernel-based Stochastic Approximation Framework for Nonlinear Operator Learningcited by 1
  29. 2025kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learningcited by 1
  30. 2026Toeplitz Based Spectral Methods for Data-driven Dynamical Systemscited by 1
  31. 2026Koopman operator theory: fundamentals, control, and applicationscited by 1
  32. 2026Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operatorscited by 1
  33. 2022Learning dynamical systems: an example from open quantum system dynamicscited by 0
  34. 2023Propagating Kernel Ambiguity Sets in Nonlinear Data-driven Dynamics Modelscited by 0
  35. 2023Koopman Learning with Episodic Memorycited by 0
  36. 2024Gaussian kernel expansion with basis functions uniformly bounded in L∞cited by 0
  37. 2025Nonparametric Sparse Online Learning of the Koopman Operatorcited by 0
  38. 2025Predicting the Energy Landscape of Stochastic Dynamical System via Physics-informed Self-supervised Learningcited by 0

50 most cited of 66 All 66 (JSON)