Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
Dynamic mode decomposition in vector-valued reproducing kernel Hilbert spaces for extracting dynamical structure among observables
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DOI：
10.1016/j.neunet.2019.04.020
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发表时间：
2018-08
期刊：
影响因子：
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通讯作者：
Keisuke Fujii;Y. Kawahara
中科院分区：
文献类型：
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作者：
Keisuke Fujii;Y. Kawahara
了解非线性动力系统在许多工程和科学领域都是具有挑战性的。动态模式分解(DMD)是一种用于Koopman算子谱分析的数值算法，作为一种不需要显式先验知识即可获得NLDS的全局模式描述的方法而引起了人们的关注。然而，由于现有的DMD算法原则上是基于标量可观测量的串联而形成的，因此它不能直接适用于具有观测数据之间的依赖结构的数据，例如，采取图序列的形式。本文建立了观测量之间具有结构的NLDS的Koopman谱分析，并提出了该问题的一种估计算法。这种方法可以从数据中提取并可视化具有可观测性结构的NLDS的底层低维全球动力学，这对于理解这类NLDS的底层动力学是有用的。为此，我们首先描述了定义在向量值再生核Hilbert空间中的Koopman算子的谱估计问题，然后通过重新定义基于张量的DMD给出了这一问题的估计方法。作为该方法的一个特例，我们提出了一种利用邻接矩阵序列进行图动力系统Koopman谱分析的数值算法--图DMD。我们通过使用合成数据和真实世界数据来研究我们方法的经验性能。
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.