Learning Koopman operators for coupled systems via information on governing equations of subsystems

Nonlinear coupled systems are ubiquitous in science and engineering. The analysis and modeling of such systems are challenging due to their high dimensionality and complex interactions among subsystems. In recent years, operator-theoretic methods based on the Koopman operator have attracted attention as a powerful tool for analyzing and modeling nonlinear dynamical systems. Extended dynamic mode decomposition (EDMD) is one of the most popular methods for approximating the Koopman operator. However, EDMD is a purely data-driven method, and it may be unstable and inaccurate for coupled systems under limited data availability. In this paper, we propose a method to construct a finite-dimensional Koopman approximation for coupled systems using the differential equations governing each subsystem. The proposed method aims to improve data efficiency by using the known subsystem dynamics as prior information and learning the coupling-induced correction from a limited number of snapshots. We also demonstrate its effectiveness through numerical experiments on coupled oscillator systems.