CompKoop: Compact Observable Completion for Projected Koopman Prediction

Observable closure presents a central challenge for long-horizon prediction with finite-dimensional Koopman models. For many nonlinear autonomous systems, a compact observable space cannot support closed linear evolution under a single finite-dimensional operator. A common approach is therefore to make the finite operator state dependent, allowing unresolved nonlinear structure to be absorbed into adaptive operator coefficients. This work investigates whether such missing structure can instead be represented through a compact learned expansion of the observable space while retaining a time-invariant projected mapping. We propose CompKoop, a Koopman framework that learns a compact product expansion of the observable space for nonlinear state prediction with a time-invariant projected readout. For state prediction, CompKoop targets the Koopman image of the physical-state observable rather than requiring closed linear evolution of the entire lifted representation, leading to a projected-closure formulation for multi-step forecasting. We further show that a class of state-dependent projected mappings can be rewritten as fixed mappings over product-expanded observables, establishing a direct connection between adaptive finite operators and observable completion. Experiments on Lorenz63 and a state-dependent Duffing system demonstrate strong long-horizon prediction with a compact parameterization. Additional experiments demonstrate accurate ten-step prediction of POD-reduced two-dimensional Navier–Stokes dynamics. Analysis of independently trained state-dependent Koopman models shows that product expansion captures their adaptive projected increments. Controlled feature probes and native-readout analysis further show that the learned product observables improve the representation of state evolution and contribute dynamical interactions to the trained predictors. These results support observable completion as a structured alternative to absorbing finite-dimensional closure deficiency through operator adaptation.