Kernelized Koopman Representation Geometry Governs Transient Amplification in Neural Dynamical Systems

We study how kernel representation geometry shapes finite-time behavior in the Koopman surrogates of neural dynamical systems. Although data-driven algorithms to reconstruct such transfer operators, in particular, Koopman operators, are common, their issue of producing transient amplification remains largely unexplored. This phenomenon is due to the kernel-induced representation geometry on the Koopman operator from the corresponding reproducing kernel Hilbert space (RKHS) and is not apparent from the spectrum alone. We quantify this framework with the help of finite-time operator growth, pesudospectral analysis, and the Kreiss constant and apply them on three neural dynamical systems. Across these systems, different kernel representations produce markedly different finite-time amplification and pseudospectral behavior despite being trained on the same underlying dynamics. Our results demonstrate the existence of kernel hierarchy in four kernels for the Koopman operator representation inside respective RKHSs, and therefore those Koopman surrogates are not representation neutral. Our analysis can serve as a practical tool for assessing kernel Koopman models of neural dynamical systems before they are employed for prediction or perturbation-response analysis.