With regards to data-driven modeling and analysis in dynamical systems, the Dynamic-Mode Decomposition (DMD) has become one of the most powerful approaches for developing deep insights into complex physical processes. In this talk, we explore novel contributions to both the descriptive and predictive aspects of DMD. With regards to modal analysis, through our development of a mode selection process, we look at how the DMD is able to identify coherent structures and relatively stable low dimensional subspaces of dynamics in weakly turbulent flows. Thus our selection method illustrates the power of the DMD to help quantify and elucidate structure in complex dispersive flows.

       With regards to prediction, we show that by merging Takens embeddings with neural networks that we are able to extend DMD to an adaptive and thereby accurate equation free modeling framework. We show that our method is able to produce good forecasts and modal decompositions across a range of chaotic dynamical systems, including a PCA reduced version of the Kuramoto-Sivashinksy equation. We likewise explore the role that information theory plays in describing the dynamics of our learning process, which hints at further improvements to our method.