Enhancing Dynamic Mode Decomposition using Autoencoder Networks

Prediction, estimation, and control of dynamical systems remain challenging due to nonlinearity. The Koopman operator is an infinite-dimensional linear operator that evolves the observables of a dynamical system which we approximate by the dynamic mode decomposition (DMD) algorithm. Using DMD to predict the evolution of a nonlinear dynamical system over extended time horizons requires choosing the right observable function defined on the state space. A number of DMD modifications have been developed to choose the right observable function, such as Extended DMD. Here, we propose a simple machine learning based approach to find these coordinate transformations. This is done via a deep autoencoder network. This simple DMD autoencoder is tested and verified on nonlinear dynamical system time series datasets, including the pendulum and fluid flow past a cylinder.

Keywords - Dynamic mode decomposition, Deep learning, Dynamical systems, Koopman analysis, Observable functions.