Description
Recent advances in computational abilities along with the vast amount of available data have led dynamical systems research away from first principles approaches and towards data driven analysis, prediction, and control. While there are many efficient procedures for the case of linear systems, and even fast and accurate techniques for nonlinear systems, these often come with the assumption that the underlying governing equations are known. In the case of unknown underlying dynamics, large datasets will often have encoded within them more information than is necessary to describe the system, in which case any number of model reduction techniques can be used. In this paper, we study the opposite case where we assume no prior knowledge of the system along with an insufficient representation of the full dynamics in a given dataset, typically requiring a higher dimensional embedding to analyze. Where previous methods required extensive testing and experimentation to find an appropriate embedding dimension, we allow an autoencoder to discover the appropriate coordinates and the necessary embedding dimension of our data using Hankel type matrices to best represent the dynamics. We refer to this method as the deep learning Hankel dynamic mode decomposition (DLHDMD). The DLHDMD was tested on an array of dynamical systems including the nonlinear pendulum, the Duffing equation, the Korteweg de Vries equation, the Van der Pol oscillator, and the chaotic Lorenz-63 system. In each of these tests the method proved to retain impressive accuracy with the advantage of a system independent neural network architecture.
