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 convolutional autoencoder networks to simultaneously find optimal families of observables which also generate both accurate embeddings of the flow into a space of observables and immersions of the observables back into flow coordinates. This network results in a global transformation of the flow and affords future state prediction via EDMD and the decoder network. We call this method 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.

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天值可能很困难。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,但是,由于用户定义的观测需要,EDMD方法面临一系列挑战。为了解决这一问题,我们探索如何使用等离子自动编码网络同时找到最佳的可观测序列,这些可同时生成准确的可观测空间流和可观测到回流坐标的探空。这一网络导致流动的全球转变,并能够通过EDMD和解密网络提供未来状态预测。我们称之为这种方法深学习的动态模式分解(DLDMD)。该方法在非线性数据集上进行测试,并显示其结果优于标准的 DMDMD方法。