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Terrestrial network limitations drive the integration of non-terrestrial networks (NTNs), notably mega-constellations comprising thousands of low Earth orbit (LEO) satellites. While these satellites act as interconnected network switches via inter-satellite links (ISLs), their massive scale creates severe bottlenecks for network management. To address this, we propose a scalable, hierarchical software-defined networking (SDN) framework. Our architecture leverages graph neural networks (GNNs) to compactly represent the constellation topology, and Koopman theory to linearize nonlinear dynamics. Specifically, a Graph Koopman Autoencoder (GKAE) forecasts spatio-temporal behavior within a linear subspace for each orbital shell. A central SDN controller then aggregates these shell-level predictions for globally coordinated control. Simulations on the Starlink constellation demonstrate that our approach achieves at least a 42.8\% improvement in spatial compression and a 10.81\% improvement in temporal forecasting compared to established baselines, all while utilizing a significantly smaller model footprint.