Oscillatory Neural Dynamics over Sheaves

2026-10-07

AI Summary

This study addresses the failure of long-range information propagation and the over-smoothing bottleneck in graph neural networks by proposing the ONDA framework. To the best of our knowledge, this work is the first to couple wave dynamics with matrix-valued transport, leveraging operator-valued information waves to facilitate long-range graph learning. By introducing second-order oscillatory dynamics alongside layered transport operators, ONDA ensures that cross-node influence remains undiminished during propagation, thereby overcoming the fundamental limitations of conventional diffusion models. Extensive experiments demonstrate that ONDA significantly outperforms existing state-of-the-art methods across diverse benchmark tasks, including long-range propagation, graph transfer learning, and heterogeneous graph modeling.

Abstract

Effective long-range propagation remains a central challenge in graph neural networks, as increasing a model's propagation depth does not guarantee that distant nodes effectively influence each other. 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 stalk-wise sensitivity analysis and show that the cross-influence never vanishes. 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.

Problem

• Graph Neural Networks
• Long-range Propagation
• Sheaf Neural Networks
• Oscillatory Dynamics

Innovation

• Sheaf Neural Networks
• Oscillatory Dynamics
• Long-range Propagation
• Operator-valued Transport
• Graph Neural Networks