论文标题
通过动态模式分解拟合结构化的非线性系统
Toward fitting structured nonlinear systems by means of dynamic mode decomposition
论文作者
论文摘要
动态模式分解（DMD）是一种数据驱动的方法，用于识别复杂非线性系统的动力学。它使用通过实验或数值模拟产生的测量时间域数据提取基本动力学的重要特征。在原始方法中，假定测量值与线性操作员大致相关。因此，将线性离散时间系统拟合到给定数据。但是，通常，非线性系统建模物理现象具有特殊的已知结构。在此贡献中，我们根据经典的DMD方法提出了一种识别和还原方法，允许将结构化的非线性系统拟合到测量数据。我们主要关注两种类型的非线性：双线性和二次双线性。通过执行这种额外的结构，可以获得更多地洞悉原始过程的非线性行为。最后，我们证明了针对不同示例的建议方法，例如汉堡方程和耦合的范德波尔振荡器。
The dynamic mode decomposition (DMD) is a data-driven method used for identifying the dynamics of complex nonlinear systems. It extracts important characteristics of the underlying dynamics using measured time-domain data produced either by means of experiments or by numerical simulations. In the original methodology, the measurements are assumed to be approximately related by a linear operator. Hence, a linear discrete-time system is fitted to the given data. However, often, nonlinear systems modeling physical phenomena have a particular known structure. In this contribution, we propose an identification and reduction method based on the classical DMD approach allowing to fit a structured nonlinear system to the measured data. We mainly focus on two types of nonlinearities: bilinear and quadratic-bilinear. By enforcing this additional structure, more insight into extracting the nonlinear behavior of the original process is gained. Finally, we demonstrate the proposed methodology for different examples, such as the Burgers' equation and the coupled van der Pol oscillators.
