Mangan N M, Askham T, Brunton S L, Kutz J N, Proctor J L
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA.
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA.
Proc Math Phys Eng Sci. 2019 Mar;475(2223):20180534. doi: 10.1098/rspa.2018.0534. Epub 2019 Mar 6.
Hybrid systems are traditionally difficult to identify and analyse using classical dynamical systems theory. Moreover, recently developed model identification methodologies largely focus on identifying a single set of governing equations solely from measurement data. In this article, we develop a new methodology, Hybrid-Sparse Identification of Nonlinear Dynamics, which identifies separate nonlinear dynamical regimes, employs information theory to manage uncertainty and characterizes switching behaviour. Specifically, we use the nonlinear geometry of data collected from a complex system to construct a set of coordinates based on measurement data and augmented variables. Clustering the data in these measurement-based coordinates enables the identification of nonlinear hybrid systems. This methodology broadly empowers nonlinear system identification without constraining the data locally in time and has direct connections to hybrid systems theory. We demonstrate the success of this method on numerical examples including a mass-spring hopping model and an infectious disease model. Characterizing complex systems that switch between dynamic behaviours is integral to overcoming modern challenges such as eradication of infectious diseases, the design of efficient legged robots and the protection of cyber infrastructures.
传统上，使用经典动力学系统理论来识别和分析混合系统是困难的。此外，最近开发的模型识别方法主要集中在仅从测量数据中识别出一组单一的控制方程。在本文中，我们开发了一种新方法——非线性动力学的混合稀疏识别，该方法可以识别不同的非线性动力学区域，运用信息论来管理不确定性并刻画切换行为。具体而言，我们利用从复杂系统收集的数据的非线性几何特性，基于测量数据和扩充变量构建一组坐标。在这些基于测量的坐标中对数据进行聚类，能够识别非线性混合系统。这种方法广泛地赋能非线性系统识别，而不会在时间上对数据进行局部约束，并且与混合系统理论有直接联系。我们在数值示例上证明了该方法的成功，这些示例包括一个质量 - 弹簧跳跃模型和一个传染病模型。刻画在动态行为之间切换的复杂系统，对于克服诸如根除传染病、设计高效的有腿机器人以及保护网络基础设施等现代挑战至关重要。
