Consistent spectral approximation of Koopman operators using resolvent compactification

Abstract

Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on $L^2$ for measure-preserving, continuous-time systems via a ``compactification'' of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.

60 pages, 7 figures. Revision includes removal of errors in notation and improved clarity

References in corpus (20)

• A Data-Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
• On Dynamic Mode Decomposition: Theory and Applications
• Chaos as an Intermittently Forced Linear System
• Consistency of spectral clustering
• Data-driven model reduction and transfer operator approximation
• Ruelle-Perron-Frobenius spectrum for Anosov maps
• On the numerical approximation of the Perron-Frobenius and Koopman operator
• Eigendecompositions of Transfer Operators in Reproducing Kernel Hilbert Spaces
• Nonparametric forecasting of low-dimensional dynamical systems
• The Lorenz attractor is mixing
• Delay-coordinate maps and the spectra of Koopman operators
• Operator-theoretic framework for forecasting nonlinear time series with kernel analog techniques
• Koopman spectra in reproducing kernel Hilbert spaces
• Reproducing kernel Hilbert space compactification of unitary evolution groups
• Bi-stochastic kernels via asymmetric affinity functions
• Koopman and Perron-Frobenius Operators on reproducing kernel Banach spaces
• Ergodic averaging with and without invariant measures
• Transfer Operators from Optimal Transport Plans for Coherent Set Detection
• On harmonic Hilbert spaces on compact abelian groups
• Data Assimilation in Operator Algebras

Cited by in corpus (2)

• Limits of Learning Dynamical Systems
• Koopman and transfer operator techniques from the perspective of quantum theory