Koopman operator theory shows how nonlinear dynamical systems can be represented as an infinite-dimensional, linear operator acting on a Hilbert space of observables of the system. However, determining the relevant modes and eigenvalues of this infinite-dimensional operator can be difficult. The extended dynamic mode decomposition (EDMD) is one such method for generating approximations to Koopman spectra and modes, but the EDMD method faces its own set of challenges due to the need of user defined observables. To address this issue, we explore the use of autoencoder networks to simultaneously find optimal families of observables which also generate both accurate embeddings of the flow into a space of observables and submersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via the EDMD and the decoder network. We call this method the deep learning dynamic mode decomposition (DLDMD). The method is tested on canonical nonlinear data sets and is shown to produce results that outperform a standard DMD approach and enable data-driven prediction where the standard DMD fails.
翻译:Koopman 操作员理论显示,非线性动态系统如何可以作为无限的、线性操作员在系统可观测的Hilbert空间上作为无限的、线性操作员。然而,确定这一无限操作员的相关模式和亚值可能是困难的。扩展的动态模式分解(EDMD)是生成Koopman光谱和模式近似的一种方法,但EDMD方法由于用户定义的观测需要而面临其自身的一系列挑战。为了解决这一问题,我们探索如何使用自动编码网络同时找到观测对象的最佳组合,从而同时生成可观测到空间的精确嵌入和可观测到流坐标上的亚值。这一网络导致流动的全球转变,并通过EDMD和解密网络提供未来状态预测。我们称这种方法为深学习动态模式分解(DLDMD) 。该方法在非线性非线性数据集上进行了测试,并显示其结果将超越标准的 DMDMD 失败情况下的驱动数据预测。