Learning Deep Neural Network Representations for Koopman Operators of Nonlinear Dynamical Systems
This addresses a bottleneck in dynamical systems analysis for applications like cyber-physical infrastructure and fluid dynamics, offering an automated alternative to manual dictionary tuning.
The paper tackles the computational complexity of Koopman operator analysis for nonlinear dynamical systems by introducing a deep learning framework that automatically selects efficient dictionaries, outperforming state-of-the-art methods and predicting 100 steps quantitatively and 400 steps qualitatively in benchmarks like the glycolytic oscillator.
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 deep learning framework for learning Koopman operators of nonlinear dynamical systems. We show that this novel method automatically selects efficient deep dictionaries, 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 quantitatively 100 steps into the future, using only a single timepoint, and qualitative oscillatory behavior 400 steps into the future.