  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.
