Oscillatory Neural Dynamics over Sheaves

Effective long-range propagation remains a central challenge in graph neural networks, as increasing model's propagation depth does not guarantee that distant information remains influential. Sheaf neural networks enrich graph propagation through matrix-valued transport between stalks, still this expressivity alone does not automatically imply effective long-range communication. We introduce ONDA, a long-range graph learning framework based on operator-valued information waves. Stalk-valued representations evolve through second-order dynamics governed by learned sheaf transport operators, combining wave-like propagation with expressive local geometry. We characterize long-range influence through a stalkwise sensitivity analysis and show that, in the conservative self-adjoint setting, the dynamics preserve node information. Across long-range propagation, severe graph bottlenecks, graph transfer, and heterophilic benchmarks, ONDA consistently improves over scalar wave propagation, diffusive sheaf baselines, and state-of-the-art models, demonstrating the benefit of coupling wave dynamics with matrix-valued transport.