mlsanityData to 7 Oct 2026

Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator

arXiv 2412.16588 ↗First posted 21 Dec 2024math.DS cs.NA math.NAcited by 5

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Abstract (arXiv)

The Koopman operator provides a linear framework to study nonlinear dynamical systems. Its spectra offer valuable insights into system dynamics, but the operator can exhibit both discrete and continuous spectra, complicating direct computations. In this paper, we introduce a kernel-based method to construct the principal eigenfunctions of the Koopman operator without explicitly computing the operator itself. These principal eigenfunctions are associated with the equilibrium dynamics, and their eigenvalues match those of the linearization of the nonlinear system at the equilibrium point. We exploit the structure of the principal eigenfunctions by decomposing them into linear and nonlinear components. The linear part corresponds to the left eigenvector of the system's linearization at the equilibrium, while the nonlinear part is obtained by solving a partial differential equation (PDE) using kernel methods. Our approach avoids common issues such as spectral pollution and spurious eigenvalues, which can arise in previous methods. We demonstrate the effectiveness of our algorithm through numerical examples.

Key facts. Cited by 5 papers in the map (data through 7 Oct 2026). Builds on 21 of its 71 references that are in the map. First posted on arXiv 21 Dec 2024.

Builds on

21 of 71 references.

References · 21

  1. 2017Deep learning for universal linear embeddings of nonlinear dynamicscited by 491
  2. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 128
  3. 2019Data-driven approximation of the Koopman generator: Model reduction, system identification, and controlcited by 85
  4. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 59
  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. 2023Koopman Kernel Regressioncited by 22
  10. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  11. 2023Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systemscited by 15
  12. 2021Learning dynamical systems from data: A simple cross-validation perspective, part III: Irregularly-Sampled Time Seriescited by 14
9 more
  1. 2020Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theoremcited by 13
  2. 2019Dimensionality Reduction of Complex Metastable Systems via Kernel Embeddings of Transition Manifoldscited by 12
  3. 2018A Note on Kernel Methods for Multiscale Systems with Critical Transitionscited by 11
  4. 2024Adversarial dynamical systems characterize when data-driven learning succeeds or failscited by 10
  5. 2019Data-driven prediction of a multi-scale Lorenz 96 chaotic system using deep learning methods: Reservoir computing, ANN, and RNN-LSTMcited by 9
  6. 2024Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spacescited by 9
  7. 2023Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dotscited by 8
  8. 2023Spectral Koopman Method for Identifying Stability Boundarycited by 5
  9. 2024Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approachcited by 4

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2025: 1; 2026: 4.

5 papers.

  1. 2026Kernel Methods for Stochastic Dynamical Systems with Application to Koopman Eigenfunctions: Feynman-Kac Representations and RKHS Approximationcited by 1
  2. 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
  3. 2025A Dynamics-Informed Gaussian Process Framework for 2D Stochastic Navier-Stokes via Quasi-Gaussianitycited by 0
  4. 2026Adaptive Kernel Selection for Kernelized Diffusion Mapscited by 0
  5. 2026Dictionary learning for Kernel EDMDcited by 0