Machine Learning Enhanced Hankel Dynamic-Mode Decomposition

While the acquisition of time series has become increasingly more straightforward and sophisticated, developing dynamical models from time series is still a challenging and ever 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 sophisticated and 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 develop an adaptive learning scheme that better captures higher dimensional and chaotic dynamics. We call this method the Deep Learning Hankel DMD (DLHDMD). We show that the DLHDMD is able to generate accurate dynamics for chaotic time series, and 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 more sophisticated deep learning methods for time series forecasting.