Learning Spectral Compositional Koopman Operators for Global-to-Regional Weather Forecasting

State-of-the-art machine learning weather prediction systems achieve strong upper-air forecast skill but degrade at longer lead times, with mesoscale structures progressively vanishing and ensemble forecasts becoming under-dispersive. While these limitations are often attributed to the training objective, we find that message-passing processors significantly contribute to this degradation. Repeated neighborhood aggregation at each autoregressive step acts as a smoothing operator that, over rollout steps, suppresses high-frequency information in latent space. This effect is particularly limiting in regional weather prediction, where fine-scale structures are already partially unresolved, further eroding mesoscale variability and hindering long-horizon forecasting. We propose GraphMet, a graph-based weather model motivated by Koopman theory that replaces stacked message-passing layers with a single linear Koopman operator step. GraphMet learns linear evolution in a spectral latent basis while preserving the encoder, decoder, and multi-scale icosahedral mesh structure. A two-term spectral objective jointly learns a stable Koopman basis and preserves latent geometry, enabling efficient long-range propagation without repeated nonlinear aggregation. Empirically, GraphMet reduces the 15-day activity and improves regional RMSE by 11-14% at 5km resolution, and enables efficient probabilistic forecasting with lower inference cost than state-of-the-art methods. Our code available at : https://anonymous.4open.science/w/GraphMet-54ED/.