Abstract
Due to the increasing complexity of dynamic systems, it is increasingly difficult for traditional mathematical methods to meet the modeling requirements of complex dynamic systems. With the continuous innovation of computer and big data technologies, massive data can be easily obtained and stored. Therefore, studies of dynamic system modeling through data-driven approaches have attracted more and more researchers’ attention. This paper compares the dynamic mode decomposition method and dynamic equation reconstruction. Taking Lorenz and nonlinear Helmholtz resonant systems as examples, the two methods show the ability to reconstruct and describe the evolution characteristics of the dynamic system. Specifically, the dynamic mode decomposition method can describe the characteristics of the dynamic system more intuitively; however, it cannot provide physical insights. On the other hand, the discovery of dynamic equations from data can more accurately express the physical evolution characteristics of the dynamic system; however, it is easily affected by random noise. Because the dynamic mode decomposition method can obtain a reduced-order model, which can not only retain useful information of the original data but also reduce the noise disturbances, it can effectively improve the noise attenuation and finally reconstructions of differential dynamical equations.
Highlights
As aeronautic and astronautic systems get more and more complex, it becomes very difficult to mathematically model complex systems from the perspective of physical mechanisms
empirical mode decomposition (EMD) decomposes scitation.org/journal/adv the signal into a sum of intrinsic mode functions (IMFs) according to the time scale characteristics without any basis function being set in advance, and the decomposed IMF components contain local characteristic signals of different time scales of the original signal
(2) Data-driven reconstruction of dynamic equations is found to have an excellent physical interpretation, but they are significantly affected by noise and are difficult to model accurately
Summary
As aeronautic and astronautic systems get more and more complex, it becomes very difficult to mathematically model complex systems from the perspective of physical mechanisms. Rudy et al. proposed a PDE functional identification of nonlinear dynamics (PDE-FIND) algorithm in 2016 This method can select the correct nonlinear term and spatial derivative term from a large library in order to identify partial differential equations. To the SINDy algorithm, a deep learning method (deep hidden physics models) was proposed to extract nonlinear partial differential equations from spatiotemporal data. This algorithm utilizes the newest developed automatic differentiation and deep neural network to learn infinite dimensional dynamic systems. In 2019, Long et al. proposed a new deep neural network called PDE-Net 2.0, which reconstructed partial differential equations from observed time-dependent dynamic data, and analyzed the underlying mechanisms that drive dynamics. The modal decomposition technology is used to construct a reducedorder model to reduce noise and reconstruct partial differential equations
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