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Published June 2025 | Published
Journal Article Metadata-only

Kernel methods for the approximation of the eigenfunctions of the Koopman operator

Lee, Jonghyeon1
Hamzi, Boumediene1, 2, 3, 4 ORCID icon
Hou, Boya5
Owhadi, Houman1 ORCID icon
Santin, Gabriele6
Vaidya, Umesh7
  • 1. ROR icon California Institute of Technology
  • 2. ROR icon The Alan Turing Institute
  • 3. ROR icon Imperial College London
  • 4. ROR icon Gulf University for Science & Technology
  • 5. ROR icon University of Illinois Urbana-Champaign
  • 6. ROR icon Ca' Foscari University of Venice
  • 7. ROR icon Clemson University

Abstract

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.

Copyright and License (English)

© 2025 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

Funding (English)

GS is a member of INdAM-GNCS, and his work was partially supported by the project “Data-driven discovery and control of multi-scale interacting artificial agent system” funded by the program Next-GenerationEU - National Recovery and Resilience Plan (NRRP) – CUP H53D23008920001. HO and JL acknowledge support from the Air Force Office of Scientific Research under MURI award number FA9550-20-1-0358 (Machine Learning and Physics-Based Modeling and Simulation) and the Department of Energy under the MMICCs SEA-CROGS award. BH acknowledges support from National Science Foundation EPCN-2031570, the Air Force Office of Scientific Research, United States (award number FA9550-21-1-0317) and the Department of Energy (award number SA22-0052-S001). HO is grateful for support from a Department of Defense Vannevar Bush Faculty Fellowship.

Contributions (English)

Jonghyeon Lee: Writing – original draft, Visualization, Software, Resources, Data curation. Boumediene Hamzi: Writing – review & editing, Writing – original draft, Methodology, Investigation. Boya Hou: Writing – original draft, Formal analysis. Houman Owhadi: Writing – review & editing, Funding acquisition. Gabriele Santin: Writing – review & editing, Formal analysis. Umesh Vaidya: Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization.

Code Availability (English)

The code for this paper can be found at https://github.com/jonghyeon1998/Koopman.

Conflict of Interest (English)

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional details

Related works Funding Dates Caltech Custom Metadata
Created:
July 15, 2025
Modified:
July 15, 2025