Kernel Learning for Data-Driven Spectral Analysis of Koopman Operators

Spectral analysis of the Koopman operators is a useful tool for studying nonlinear dynamical systems and has been utilized in various branches of science and engineering for purposes such as understanding complex phenomena and designing a controller. Several methods to compute the Koopman spectral analysis have been studied, among which data-driven methods are attracting attention. We focus on one of the popular data-driven methods, which is based on the Galerkin approximation of the operator using a basis estimated in a data-driven manner via the diffusion maps algorithm. The performance of this method with a finite amount of data depends on the choice of the kernel function used in diffusion maps, which creates a need for kernel selection. In this paper, we propose a method to learn the kernel function adaptively to obtain better performance in approximating spectra of the Koopman operator using the Galerkin approximation with diffusion maps. The proposed method depends on the multiple kernel learning scheme, and our objective function is based on the idea that a diffusion operator should commute with the Koopman operator. We also show the effectiveness of the proposed method empirically with numerical examples.