Kernel Methods for the Approximation of the Eigenfunctions of the Koopman Operator
arXiv:2412.16588 · doi:10.1016/j.physd.2025.134662
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.
References in corpus (24)
- Discovering governing equations from data: Sparse identification of nonlinear dynamical systems
- A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
- Deep learning for universal linear embeddings of nonlinear dynamics
- Applied Koopmanism
- Using Machine Learning to Replicate Chaotic Attractors and Calculate Lyapunov Exponents from Data
- Predicting catastrophes in nonlinear dynamical systems by compressive sensing
- Data-driven approximation of the Koopman generator: Model reduction, system identification, and control
- Kernel Flows: from learning kernels from data into the abyss
- Physics-constrained, low-dimensional models for MHD: First-principles and data-driven approaches
- Learning dominant physical processes with data-driven balance models
- Learning dynamical systems from data: a simple cross-validation perspective
- Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
- Kernel-based approximation of the Koopman generator and Schrödinger operator
- Operator-valued Kernels for Learning from Functional Response Data
- Koopman spectra in reproducing kernel Hilbert spaces
- Kernel methods for center manifold approximation and a data-based version of the Center Manifold Theorem
- A Note on Kernel Methods for Multiscale Systems with Critical Transitions
- Greedy Kernel Methods for Center Manifold Approximation
- Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems
- Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots
- Koopman Kernel Regression
- Koopman operators with intrinsic observables in rigged reproducing kernel Hilbert spaces
- Gaussian Processes simplify differential equations
- Kernel Sum of Squares for Data Adapted Kernel Learning of Dynamical Systems from Data: A global optimization approach