Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems

The Koopman operator has recently garnered much attention for its value in dynamical systems analysis and data-driven model discovery. However, its application has been hindered by the computational complexity of extended dynamic mode decomposition; this requires a combinatorially large basis set to adequately describe many nonlinear systems of interest, e.g. cyber-physical infrastructure systems, biological networks, social systems, and fluid dynamics. Often the dictionaries generated for these problems are manually curated, requiring domain-specific knowledge and painstaking tuning. In this paper we introduce a computational framework for learning Koopman operators of nonlinear dynamical systems using deep learning. We show that this novel method automatically selects efficient deep dictionaries, requiring much lower dimensional dictionaries while outperforming state-of-the-art methods. We benchmark this method on partially observed nonlinear systems, including the glycolytic oscillator and show it is able to predict on test data quantitatively 100 steps into the future, using only a single timepoint as an initial condition, and quantitative oscillatory behavior 400 steps into the future.