每一个可逆的保测度动力系统都包含一个Koopman算子，它是一个线性的酉演化算子，作用于与不变测度相关的观测量的$L^2$空间。Koopman本征函数表示动力学的准周期或非混合分量。从给定的时间序列中提取这些本征函数及其相关的本征频率是一个重要的问题，当底层系统具有密集的点谱，或连续谱行为类似于噪声。本文介绍了识别Koopman特征频率和特征函数的方法，从离散采样的时间序列所产生的这样一个系统与未知的动态。我们的主要结果给出了傅立叶函数的充要条件，定义在$N$状态采样沿轨道的动态，可扩展到Koopman特征函数的整个状态空间，躺在再生核希尔伯特空间（RKHS）。特别是，我们表明，这样的扩展存在当且仅当傅立叶函数的RKHS范数不发散为$ N \to \infty $，在这种情况下，相应的傅立叶频率也是库普曼本征频率。如果这样的RKHS扩展不存在，我们仍然可以构造本征函数的$L^2$近似。对具有弱周期分量的混合谱系统的数值实验表明，与传统的基于离散傅里叶变换的谱估计方法相比，该方法在识别Koopman特征频率方面具有更高的技巧.
英文摘要：
Every invertible, measure-preserving dynamical system induces a Koopman operator, which is a linear, unitary evolution operator acting on the $L^2$ space of observables associated with the invariant measure. Koopman eigenfunctions represent the quasiperiodic, or non-mixing, component of the dynamics. The extraction of these eigenfunctions and their associated eigenfrequencies from a given time series is a non-trivial problem when the underlying system has a dense point spectrum, or a continuous spectrum behaving similarly to noise. This paper describes methods for identifying Koopman eigenfrequencies and eigenfunctions from a discretely sampled time series generated by such a system with unknown dynamics. Our main result gives necessary and sufficient conditions for a Fourier function, defined on $N$ states sampled along an orbit of the dynamics, to be extensible to a Koopman eigenfunction on the whole state space, lying in a reproducing kernel Hilbert space (RKHS). In particular, we show that such an extension exists if and only if the RKHS norm of the Fourier function does not diverge as $ N \to \infty $, in which case the corresponding Fourier frequency is also a Koopman eigenfrequency. If such an RKHS extension does not exist, we can still construct an $L^2$ approximation of the eigenfunction. Numerical experiments on mixed-spectrum systems with weak periodic components demonstrate that this approach has significantly higher skill in identifying Koopman eigenfrequencies compared to conventional spectral estimation techniques based on the discrete Fourier transform.
Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces
Istituto Italiano di Tecnologia（意大利理工学院）；University of Novi Sad（诺维萨德大学）；University College London（伦敦大学学院）；University of Genova（热那亚大学）；Massachusetts Institute of Technology（麻省理工学院）
Kernel Methods for the Construction of Certified Lyapunov Functions via Approximate Koopman Eigenfunctions
University of Sussex（萨塞克斯大学）；University of Iceland（冰岛大学）；Caltech（加州理工学院）；Alan Turing Institute（艾伦·图灵研究所）；Ca’ Foscari University of Venice（威尼斯卡福斯卡里大学）；Clemson University（克莱姆森大学）
Kernel 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 Characteristics
Department of Computing and Mathematical Sciences, Caltech（加州理工学院计算与数学科学系）；The Alan Turing Institute（艾伦·图灵研究所）；Department of Mechanical Engineering, Clemson University（克莱姆森大学机械工程系）
Verifiable Regularity Criterion for Conditional Expectation Operators and Conditional Mean Embeddings with Applications to Nonparametric Regression, Bayesian Inverse Problems, and Koopman Operators
Technische Universität Ilmenau（伊尔梅瑙工业大学）；Freie Universität Berlin（柏林自由大学）；Chemnitz University of Technology（开姆尼茨工业大学）
