Dylewsky Daniel, Kaiser Eurika, Brunton Steven L, Kutz J Nathan
Department of Physics, University of Washington, Seattle, Washington 98195, USA.
Department of Mechanical Engineering, University of Washington, Seattle, Washington 98195, USA.
Phys Rev E. 2022 Jan;105(1-2):015312. doi: 10.1103/PhysRevE.105.015312.
Delay embeddings of time series data have emerged as a promising coordinate basis for data-driven estimation of the Koopman operator, which seeks a linear representation for observed nonlinear dynamics. Recent work has demonstrated the efficacy of dynamic mode decomposition (DMD) for obtaining finite-dimensional Koopman approximations in delay coordinates. In this paper we demonstrate how nonlinear dynamics with sparse Fourier spectra can be (i) represented by a superposition of principal component trajectories and (ii) modeled by DMD in this coordinate space. For continuous or mixed (discrete and continuous) spectra, DMD can be augmented with an external forcing term. We present a method for learning linear control models in delay coordinates while simultaneously discovering the corresponding exogenous forcing signal in a fully unsupervised manner. This extends the existing DMD with control (DMDc) algorithm to cases where a control signal is not known a priori. We provide examples to validate the learned forcing against a known ground truth and illustrate their statistical similarity. Finally, we offer a demonstration of this method applied to real-world power grid load data to show its utility for diagnostics and interpretation on systems in which somewhat periodic behavior is strongly forced by unknown and unmeasurable environmental variables.
时间序列数据的延迟嵌入已成为一种很有前景的坐标基础，用于数据驱动的柯普曼算子估计，该算子寻求观测到的非线性动力学的线性表示。最近的工作证明了动态模态分解（DMD）在延迟坐标中获得有限维柯普曼近似的有效性。在本文中，我们展示了具有稀疏傅里叶谱的非线性动力学如何能够（i）由主成分轨迹的叠加来表示，以及（ii）在该坐标空间中由DMD进行建模。对于连续或混合（离散和连续）谱，DMD可以通过一个外部强迫项进行扩充。我们提出了一种在延迟坐标中学习线性控制模型的方法，同时以完全无监督的方式发现相应的外生强迫信号。这将现有的带控制的DMD（DMDc）算法扩展到了事先不知道控制信号的情况。我们提供了一些例子，以针对已知的真实情况验证所学习的强迫，并说明它们的统计相似性。最后，我们展示了将该方法应用于实际电网负荷数据的情况，以显示其在对由未知且不可测量环境变量强烈强迫呈现某种周期性行为的系统进行诊断和解释方面的实用性。
