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Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces

arXiv 2509.22482 ↗First posted 26 Sep 2025cs.LG math.DS nlin.CDcited by 0

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Abstract (arXiv)

The Koopman operator, as a linear representation of a nonlinear dynamical system, has been attracting attention in many fields of science. Recently, Koopman operator theory has been combined with another concept that is popular in data science: reproducing kernel Hilbert spaces. We follow this thread into Gaussian process methods, and illustrate how these methods can alleviate two pervasive problems with kernel-based Koopman algorithms. The first being sparsity: most kernel methods do not scale well and require an approximation to become practical. We show that not only can the computational demands be reduced, but also demonstrate improved resilience against sensor noise. The second problem involves hyperparameter optimization and dictionary learning to adapt the model to the dynamical system. In summary, the main contribution of this work is the unification of Gaussian process regression and dynamic mode decomposition.

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35 of 143 references.

References · 35

  1. 2016Kernel Mean Embedding of Distributions: A Review and Beyondcited by 364
  2. 2017On Kernelized Multi-armed Banditscited by 352
  3. 2021Modern Koopman Theory for Dynamical Systemscited by 276
  4. 2017Learning Koopman Invariant Subspaces for Dynamic Mode Decompositioncited by 158
  5. 2017VAMPnets: Deep learning of molecular kineticscited by 136
  6. 2014Dynamic mode decomposition with controlcited by 129
  7. 2017Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systemscited by 127
  8. 2020Forecasting Sequential Data using Consistent Koopman Autoencoderscited by 115
  9. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  10. 2017Time-lagged autoencoders: Deep learning of slow collective variables for molecular kineticscited by 65
  11. 2019Rates of Convergence for Sparse Variational Gaussian Process Regressioncited by 63
  12. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 60
23 more
  1. 2015On Sparse variational methods and the Kullback-Leibler divergence between stochastic processescited by 57
  2. 2017Variational approach for learning Markov processes from time series datacited by 52
  3. 2017Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spacescited by 48
  4. 2021Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systemscited by 45
  5. 2021Reproducing Kernel Hilbert Space, Mercer's Theorem, Eigenfunctions, Nystr"om Method, and Use of Kernels in Machine Learning: Tutorial and Surveycited by 37
  6. 2022Residual Dynamic Mode Decomposition: Robust and verified Koopmanismcited by 36
  7. 2017Koopman operator-based model reduction for switched-system control of PDEscited by 31
  8. 2020Kernel-based approximation of the Koopman generator and Schr"odinger operatorcited by 31
  9. 2023The Multiverse of Dynamic Mode Decomposition Algorithmscited by 26
  10. 2019Modeling and Control of Soft Robots Using the Koopman Operator and Model Predictive Controlcited by 23
  11. 2021Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)cited by 22
  12. 2023Koopman Kernel Regressioncited by 22
  13. 2020Data-Driven Model Predictive Control using Interpolated Koopman Generatorscited by 17
  14. 2019Kernel methods for detecting coherent structures in dynamical datacited by 13
  15. 2018Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observablescited by 12
  16. 2023Beyond expectations: Residual Dynamic Mode Decomposition and Variance for Stochastic Dynamical Systemscited by 11
  17. 2020Sparsity-promoting algorithms for the discovery of informative Koopman invariant subspacescited by 10
  18. 2018A kernel-based approach to molecular conformation analysiscited by 8
  19. 2020Value Function Approximations via Kernel Embeddings for No-Regret Reinforcement Learningcited by 7
  20. 2025Koopman-Equivariant Gaussian Processescited by 6
  21. 2023Reprojection methods for Koopman-based modelling and predictioncited by 5
  22. 2024Learning dynamical systems from data: Gradient-based dictionary optimizationcited by 3
  23. 2023Avoiding discretization issues for nonlinear eigenvalue problemscited by 1

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