The kernel perspective on dynamic mode decomposition
This work addresses foundational theoretical gaps in DMD for researchers in dynamical systems and machine learning, offering a more general framework but is incremental in refining existing methods.
The paper tackles theoretical limitations in dynamic mode decomposition (DMD) by providing counterexamples to common assumptions about Koopman operators, such as boundedness and eigenfunction existence, and proves that Gaussian RBF kernel spaces only support bounded operators for affine dynamics. It introduces a new DMD framework requiring only densely defined operators over RKHSs, demonstrating effectiveness with numerical examples.
This manuscript revisits theoretical assumptions concerning dynamic mode decomposition (DMD) of Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators. Counterexamples that illustrate restrictiveness of the assumptions are provided for each of the assumptions. In particular, this manuscript proves that the native reproducing kernel Hilbert space (RKHS) of the Gaussian RBF kernel function only supports bounded Koopman operators if the dynamics are affine. In addition, a new framework for DMD, that requires only densely defined Koopman operators over RKHSs is introduced, and its effectiveness is demonstrated through numerical examples.