While the acquisition of time series has become more straightforward, developing dynamical models from time series is still a challenging and evolving problem domain. Within the last several years, to address this problem, there has been a merging of machine learning tools with what is called the dynamic mode decomposition (DMD). This general approach has been shown to be an especially promising avenue for accurate model development. Building on this prior body of work, we develop a deep learning DMD based method which makes use of the fundamental insight of Takens' Embedding Theorem to build an adaptive learning scheme that better approximates higher dimensional and chaotic dynamics. We call this method the Deep Learning Hankel DMD (DLHDMD). We likewise explore how our method learns mappings which tend, after successful training, to significantly change the mutual information between dimensions in the dynamics. This appears to be a key feature in enhancing the DMD overall, and it should help provide further insight for developing other deep learning methods for time series analysis and model generation.

翻译：虽然时间序列的获取已变得更加便捷，但从时间序列中建立动力学模型仍是一个具有挑战性且不断演进的领域。近年来，为解决这一问题，机器学习工具与所谓动态模态分解（DMD）出现了融合趋势。这种通用方法已被证明是精确模型开发的一条极具前景的路径。基于先前的研究工作，我们提出了一种基于深度学习的DMD方法，该方法利用Takens嵌入定理的基本洞察，构建了一种自适应学习方案，能够更好地逼近高维和混沌动力学。我们将此方法称为深度学习汉克尔DMD（DLHDMD）。我们还探讨了我们的方法如何学习映射，这些映射在成功训练后往往显著改变动力学维度之间的互信息。这似乎是整体增强DMD的关键特征，并将有助于为时间序列分析和模型生成开发其他深度学习方法提供进一步的见解。