Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables

Understanding nonlinear dynamical systems (NLDSs) is challenging in a variety of engineering and scientific fields. Dynamic mode decomposition (DMD), which is a numerical algorithm for the spectral analysis of Koopman operators, has been attracting attention as a way of obtaining global modal descriptions of NLDSs without requiring explicit prior knowledge. However, since existing DMD algorithms are in principle formulated based on the concatenation of scalar observables, it is not directly applicable to data with dependent structures among observables, which take, for example, the form of a sequence of graphs. In this paper, we formulate Koopman spectral analysis for NLDSs with structures among observables and propose an estimation algorithm for this problem. This method can extract and visualize the underlying low-dimensional global dynamics of NLDSs with structures among observables from data, which can be useful in understanding the underlying dynamics of such NLDSs. To this end, we first formulate the problem of estimating spectra of the Koopman operator defined in vector-valued reproducing kernel Hilbert spaces, and then develop an estimation procedure for this problem by reformulating tensor-based DMD. As a special case of our method, we propose the method named as Graph DMD, which is a numerical algorithm for Koopman spectral analysis of graph dynamical systems, using a sequence of adjacency matrices. We investigate the empirical performance of our method by using synthetic and real-world data.Comment: 34 pages with 4 figures, Published in Neural Networks, 201

Operator-Based Detecting, Learning, and Stabilizing Unstable Periodic Orbits of Chaotic Attractors

This paper examines the use of operator-theoretic approaches to the analysis of chaotic systems through the lens of their unstable periodic orbits (UPOs). Our approach involves three data-driven steps for detecting, identifying, and stabilizing UPOs. We demonstrate the use of kernel integral operators within delay coordinates as an innovative method for UPO detection. For identifying the dynamic behavior associated with each individual UPO, we utilize the Koopman operator to present the dynamics as linear equations in the space of Koopman eigenfunctions. This allows for characterizing the chaotic attractor by investigating its principal dynamical modes across varying UPOs. We extend this methodology into an interpretable machine learning framework aimed at stabilizing strange attractors on their UPOs. To illustrate the efficacy of our approach, we apply it to the Lorenz attractor as a case study.Comment: arXiv admin note: text overlap with arXiv:2304.0783

Kernel Analog Forecasting: Multiscale Test Problems

Data-driven prediction is becoming increasingly widespread as the volume of data available grows and as algorithmic development matches this growth. The nature of the predictions made, and the manner in which they should be interpreted, depends crucially on the extent to which the variables chosen for prediction are Markovian, or approximately Markovian. Multiscale systems provide a framework in which this issue can be analyzed. In this work kernel analog forecasting methods are studied from the perspective of data generated by multiscale dynamical systems. The problems chosen exhibit a variety of different Markovian closures, using both averaging and homogenization; furthermore, settings where scale-separation is not present and the predicted variables are non-Markovian, are also considered. The studies provide guidance for the interpretation of data-driven prediction methods when used in practice.Comment: 30 pages, 14 figures; clarified several ambiguous parts, added references, and a comparison with Lorenz' original method (Sec. 4.5