Monitoring the networked dynamics via the subset of nodes is essential for a variety of scientific and operational purposes. When there is a lack of an explicit model and networked signal space, traditional observability analysis and non-convex methods are insufficient. Current data-driven Koopman linearization, although derives a linear evolution model for selected vector-valued observable of original state-space, may result in a large sampling set due to: (i) the large size of polynomial based observables ($O(N^2)$, $N$ number of nodes in network), and (ii) not factoring in the nonlinear dependency betweenobservables. In this work, to achieve linear scaling ($O(N)$) and a small set of sampling nodes, wepropose to combine a novel Log-Koopman operator and nonlinear Graph Fourier Transform (NL-GFT) scheme. First, the Log-Koopman operator is able to reduce the size of observables by transforming multiplicative poly-observable to logarithm summation. Second, anonlinear GFT concept and sampling theory are provided to exploit the nonlinear dependence of observables for observability analysis using Koopman evolution model. The results demonstrate that the proposed Log-Koopman NL-GFT scheme can (i) linearize unknownnonlinear dynamics using $O(N)$ observables, and (ii) achieve lower number of sampling nodes, compared with the state-of-the art polynomial Koopman based observability analysis.

中文翻译：

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使用 Log-Koopman 非线性图傅立叶变换对网络动力学进行采样和推理

通过节点子集监控网络动态对于各种科学和操作目的至关重要。当缺乏显式模型和网络化信号空间时，传统的可观测性分析和非凸方法是不够的。当前数据驱动的 Koopman 线性化虽然为原始状态空间的选定向量值可观察量导出了线性演化模型，但由于以下原因可能会导致大样本集：（i）基于多项式的大尺寸可观察量（$O(N^2)$, $N$网络中的节点数），以及（ii）不考虑可观测值之间的非线性依赖性。在这项工作中，为了实现线性缩放（$O(N)$) 和一小组采样节点，我们建议结合新的 Log-Koopman 算子和非线性图傅立叶变换 (NL-GFT) 方案。首先，Log-Koopman 算子能够通过将乘法 poly-observable 转换为对数求和来减小 observable 的大小。其次，提供了非线性 GFT 概念和采样理论，以利用可观测值的非线性依赖性进行使用 Koopman 演化模型的可观测性分析。结果表明，所提出的 Log-Koopman NL-GFT 方案可以 (i) 使用$O(N)$ 与最先进的基于多项式 Koopman 的可观察性分析相比，可观察量，以及 (ii) 实现较少数量的采样节点。