Dynamic Mode Decomposition for Continuous Time Systems with the Liouville Operator
Dynamic Mode Decomposition for Continuous Time Systems with the Liouville Operator
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DOI：
10.1007/s00332-021-09746-w
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发表时间：
2019-10
影响因子：
3
通讯作者：
Joel A. Rosenfeld;R. Kamalapurkar;L. Gruss;Taylor T. Johnson
中科院分区：
文献类型：
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作者：
Joel A. Rosenfeld;R. Kamalapurkar;L. Gruss;Taylor T. Johnson
动态模式分解 (DMD) 已成为库普曼算子的代名词，其中使用库普曼（即合成）算子对连续时间动态进行离散化和检查。使用新引入的“占用内核”，本手稿开发了一种直接通过刘维尔算子处理连续时间动态的 DMD 方法。本手稿概述了用于离散时间系统的基于库普曼的 DMD 和用于连续时间系统的基于刘维尔的 DMD 之间的技术和理论差异，其中包括对几个再生核希尔伯特空间上的库普曼和刘维尔算子的检查。虽然刘维尔算子是模态无界的，但本手稿引入了缩放刘维尔算子的概念，对于许多动力系统来说，它是指数点积核的本机空间上的紧凑算子。缩放Liouville算子的紧凑性允许基于Liouville的DMD的范数收敛，这是相对于基于Koopman的DMD的决定性优势。
Dynamic mode decomposition (DMD) has become synonymous with the Koopman operator, where continuous time dynamics are discretized and examined using Koopman (i.e. composition) operators. Using the newly introduced “occupation kernels,” the present manuscript develops an approach to DMD that treats continuous time dynamics directly through the Liouville operator. This manuscript outlines the technical and theoretical differences between Koopman-based DMD for discrete time systems and Liouville-based DMD for continuous time systems, which includes an examination of Koopman and Liouville operators over several reproducing kernel Hilbert spaces. While Liouville operators are modally unbounded, this manuscript introduces the concept of a scaled Liouville operator, which, for many dynamical systems, is a compact operator over the native space of the exponential dot product kernel. Compactness of scaled Liouville operators allows for norm convergence of Liouville-based DMD, which is a decided advantage over Koopman-based DMD.