用小波构造新观测量，提升动态模式分解对复杂系统的分析能力
Wavelet-Based Observables for Koopman Analysis: An Extended Dynamic Mode Decomposition Framework
- 基于小波变换构建柯普曼半群的特征函数作为观测量
- 理论证明小波观测量在紧致集上满足柯普曼算子的本征性质
- 提出cWDMD算法，适用于非线性系统动态行为建模
我们通过小波变换对柯普曼半群进行了深入分析。首先引入基于小波的观测量，并证明当该半群定义在紧致前向不变集上的连续函数空间（带一致范数）时，这些观测量是柯普曼半群的特征函数。随后，我们推导出柯普曼半群及其预解算子作用的闭式表达式。为实现数值逼近，我们将扩展动态模式分解（EDMD）与所提出的小波观测量结合，得到基于连续小波变换的波形动态模式分解（cWDMD）算法。我们在两个数值例子中验证了理论结果的有效性。
原文摘要 · Abstract (English)
We present an in-depth analysis of the Koopman semigroup via wavelet transform. Towards this goal, we start by introducing the wavelet-based observables and show that they are eigenfunctions of the Koopman semigroup when this semigroup is considered over the Banach space of continuous functions on a compact forward-invariant set endowed with the supremum norm. We then construct closed-form expressions of the action of the Koopman semigroup and its resolvent in terms of these observables. To approximate the action of Koopman semigroup numerically, we combine Extended Dynamic Mode Decomposition (EDMD) with the proposed wavelet-based observables leading to the Wavelet Dynamic Mode Decomposition via Continuous Wavelet Transform (cWDMD) algorithm. We validate our theoretical results on two numerical examples.
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