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.
翻译:动态模式分解(DMD)是一种数据驱动方法,用于确定复杂的非线性系统的动态。它利用通过实验或数字模拟产生的测量时间范围数据,提取了基础动态的重要特征。在最初的方法中,测量假定线性操作者大致相关。因此,线性离散时间系统与给定数据相适应。然而,非线性系统模拟物理现象往往有一个特定的已知结构。在此贡献中,我们建议了一种基于传统DMD方法的识别和减少方法,使结构化的非线性系统适应测量的数据。我们主要侧重于两类非线性非线性数据:双线和二次线性比线性比线性比线性。通过执行这一额外结构,可以更多地了解原始过程的非线性行为。最后,我们展示了不同实例的拟议方法,如Burgers的方程式和配合的van der Pol振荡器。