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

TL;DR

This paper introduces a deep learning approach to efficiently learn Koopman operators for nonlinear dynamical systems, automating dictionary selection and outperforming existing methods in long-term prediction tasks.

Contribution

It presents a novel deep learning framework that automatically learns effective dictionaries for Koopman operators, reducing manual tuning and computational complexity.

Findings

Outperforms state-of-the-art methods in benchmark tests

Accurately predicts 100 future steps from a single initial point

Captures qualitative oscillations 400 steps ahead

Abstract

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.