Koopman and Perron–Frobenius operators on reproducing kernel Banach spaces

Ikeda, Masahiro; Ishikawa, Isao; Schlosser, Corbinian

doi:10.1063/5.0094889

Cited by 8 publications

(5 citation statements)

References 33 publications

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“…The methods outlined above perform approximation of Koopman operators on L p spaces, oftentimes associated with invariant measures. In recent years, a distinct line of work has emerged that focuses on approximation of Koopman operators on RKHSs [1,24,26,42,46,49,54,64,65]. Unlike L 2 spaces whose elements are equivalence classes of functions, RKHSs are function spaces with continuous pointwise evaluation functionals.…”

Section: Review Of Data-driven Methodologiesmentioning

confidence: 99%

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Consistent spectral approximation of Koopman operators using resolvent compactification

Giannakis,

Valva

2024

Nonlinearity

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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.

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Section: Review Of Data-driven Methodologiesmentioning

confidence: 99%

“…At a minimum, the kernel should be chosen such that the Koopman operator on H is densely defined and closable. Provided that such a kernel can be found (which is, in general, non-trivial [42]), the Koopman operator on H will typically be unbounded, introducing theoretical and numerical complications.…”

Section: Review Of Data-driven Methodologiesmentioning

confidence: 99%

Consistent spectral approximation of Koopman operators using resolvent compactification

Giannakis,

Valva

2024

Nonlinearity

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“…Ideally, one should choose the kernel so that the Koopman operator on the RKHS is not only densely defined but also closable. Finding such a kernel is generally non-trivial, as indicated in (Ikeda et al 2022).…”

Section: Choices Of Dictionarymentioning

confidence: 99%

Residual dynamic mode decomposition: robust and verified Koopmanism

2023

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Dynamic mode decomposition (DMD) describes complex dynamic processes through a hierarchy of simpler coherent features. DMD is regularly used to understand the fundamental characteristics of turbulence and is closely related to Koopman operators. However, verifying the decomposition, equivalently the computed spectral features of Koopman operators, remains a significant challenge due to the infinite-dimensional nature of Koopman operators. Challenges include spurious (unphysical) modes and dealing with continuous spectra, which both occur regularly in turbulent flows. Residual dynamic mode decomposition (ResDMD), introduced by Colbrook & Townsend (Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems. 2021. arXiv:2111.14889), overcomes such challenges through the data-driven computation of residuals associated with the full infinite-dimensional Koopman operator. ResDMD computes spectra and pseudospectra of general Koopman operators with error control and computes smoothed approximations of spectral measures (including continuous spectra) with explicit high-order convergence theorems. ResDMD thus provides robust and verified Koopmanism. We implement ResDMD and demonstrate its application in various fluid dynamic situations at varying Reynolds numbers from both numerical and experimental data. Examples include vortex shedding behind a cylinder, hot-wire data acquired in a turbulent boundary layer, particle image velocimetry data focusing on a wall-jet flow and laser-induced plasma acoustic pressure signals. We present some advantages of ResDMD: the ability to resolve nonlinear and transient modes verifiably; the verification of learnt dictionaries; the verification of Koopman mode decompositions; and spectral calculations with reduced broadening effects. We also discuss how a new ordering of modes via residuals enables greater accuracy than the traditional modulus ordering (e.g. when forecasting) with a smaller dictionary. This result paves the way for more significant dynamic compression of large datasets without sacrificing accuracy.

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“…With continuous eigenfunctions for continuous systems proved valid in [47,Lemma 5.1], [23,Theorem 1], the space of continuous functions over a compact set is naturally the space of interest. On a non-recurrent domain, there exist uniquely defined non-trivial eigenfunctions and, by [70, Theorem 3.0.2], the spectrum is rich -with any eigenvalue in the closed complex unit disk legitimate [71]. Further, by [23,Theorem 2], this richness is inherited by the Koopman eigenfunctions -making them universal approximators of continuous functions.…”

Section: Supplementary Materialsmentioning

confidence: 99%

Diffeomorphically Learning Stable Koopman Operators

Bevanda

Beier

Sebastian

et al. 2022

IEEE Control Syst. Lett.

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Many machine learning approaches for decision making, such as reinforcement learning, rely on simulators or predictive models to forecast the time-evolution of quantities of interest, e.g., the state of an agent or the reward of a policy. Forecasts of such complex phenomena are commonly described by highly nonlinear dynamical systems, making their use in optimization-based decision-making challenging. Koopman operator theory offers a beneficial paradigm for addressing this problem by characterizing forecasts via linear dynamical systems. This makes system analysis and long-term predictions simple -involving only matrix multiplications. However, the transformation to a linear system is generally non-trivial and unknown, requiring learning-based approaches. While there exists a variety of approaches, they usually lack crucial learning-theoretic guarantees, such that the behavior of the obtained models with increasing data and dimensionality is often unclear. We address the aforementioned by deriving a novel reproducing kernel Hilbert space (RKHS) that solely spans transformations into linear dynamical systems. The resulting Koopman Kernel Regression (KKR) framework enables the use of statistical learning tools from function approximation for novel convergence results and generalization risk bounds under weaker assumptions than existing work. Our numerical experiments indicate advantages over state-of-the-art statistical learning approaches for Koopman-based predictors.

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Koopman and Perron–Frobenius operators on reproducing kernel Banach spaces

Cited by 8 publications

References 33 publications

Consistent spectral approximation of Koopman operators using resolvent compactification

Consistent spectral approximation of Koopman operators using resolvent compactification

Residual dynamic mode decomposition: robust and verified Koopmanism

Diffeomorphically Learning Stable Koopman Operators

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