Physics-aware kernel Koopman operator estimation for consistent nonlinear mode decomposition

Nonlinear dynamical systems are ubiquitous across scientific disciplines, yet their analysis and predictive modeling remain challenging due to their inherent complexity. Koopman operator estimation and Koopman mode decomposition are common tools for emulating and extracting modes of variability from such systems. In this work, we propose a novel method for Koopman operator estimation called the Physics-Aware Koopman Operator (PAKO). Our approach is tailored for physical consistency by introducing a regularization term based on the Hilbert-Schmidt Independence Criterion (HSIC) to enforce independence between predictions and sensitive or protected physical variables. In addition to Koopman operator estimation, we extract Koopman modes and eigenvalues through a Koopman mode decomposition. We validate PAKO on the ClimateBench dataset, demonstrating superior accuracy, robustness, and interpretability for estimating the internal variability of climate systems. Our results showcase the potential of PAKO for advancing Koopman operator estimation of complex nonlinear dynamical systems.