Hirsh Seth M, Ichinaga Sara M, Brunton Steven L, Nathan Kutz J, Brunton Bingni W
Department of Physics, University of Washington, Seattle, WA USA.
Department of Applied Mathematics, University of Washington, Seattle, WA USA.
Proc Math Phys Eng Sci. 2021 Oct;477(2254):20210097. doi: 10.1098/rspa.2021.0097. Epub 2021 Oct 13.
Time-delay embedding and dimensionality reduction are powerful techniques for discovering effective coordinate systems to represent the dynamics of physical systems. Recently, it has been shown that models identified by dynamic mode decomposition on time-delay coordinates provide linear representations of strongly nonlinear systems, in the so-called Hankel alternative view of Koopman (HAVOK) approach. Curiously, the resulting linear model has a matrix representation that is approximately antisymmetric and tridiagonal; for chaotic systems, there is an additional forcing term in the last component. In this paper, we establish a new theoretical connection between HAVOK and the Frenet-Serret frame from differential geometry, and also develop an improved algorithm to identify more stable and accurate models from less data. In particular, we show that the sub- and super-diagonal entries of the linear model correspond to the intrinsic curvatures in the Frenet-Serret frame. Based on this connection, we modify the algorithm to promote this antisymmetric structure, even in the noisy, low-data limit. We demonstrate this improved modelling procedure on data from several nonlinear synthetic and real-world examples.
时间延迟嵌入和降维是用于发现有效坐标系以表示物理系统动力学的强大技术。最近，研究表明，在时间延迟坐标上通过动态模式分解识别的模型在所谓的库普曼汉克尔替代视图（HAVOK）方法中为强非线性系统提供了线性表示。奇怪的是，所得的线性模型具有近似反对称和三对角的矩阵表示；对于混沌系统，最后一个分量中有一个额外的强迫项。在本文中，我们在HAVOK和微分几何中的弗伦内 - 塞雷标架之间建立了一种新的理论联系，并且还开发了一种改进算法，以便从更少的数据中识别出更稳定、更准确的模型。特别是，我们表明线性模型的次对角线和超对角线元素对应于弗伦内 - 塞雷标架中的内在曲率。基于这种联系，我们修改算法以促进这种反对称结构，即使在有噪声、低数据的情况下也是如此。我们在几个非线性合成和实际示例的数据上展示了这种改进的建模过程。
