Scalable Infinitesimal Generator–Based Koopman Learning

Minseok Jeong, Hyo-Sang Shin, SooJean Han

Abstract:

Koopman operator theory offers a linear representation of nonlinear dynamics and has strong potential for long-horizon prediction. However, most existing methods rely on one-step snapshot fitting and suffer from error accumulation over long rollouts. Prior work has attempted to address this by extending training horizons or using physics-informed, generator-based formulations. Still, these approaches remain limited, partly because they rely on MLP-based observables, whose spectral bias favors low-frequency components.

In this paper, we introduce a Random Fourier Feature–lifted physics-informed Koopman network (RFF-PIKN) that directly minimizes the generator loss. We first show that snapshot-based local transition fitting has inherent limitations in long-horizon stability relative to generator-based learning. We then prove that RFF-PIKN converges stably to a local minimizer of the true population risk under the generator loss. Finally, empirical comparisons demonstrate that RFF-PIKN outperforms MLP- and polynomial-based observables in long-horizon prediction while substantially reducing computation, and further matches key behaviors of oracle kernel-based methods.