从扩散到反应扩散：超图神经网络中过平滑的动力系统视角
From Diffusion to Reaction-Diffusion: A Dynamical-Systems View of Oversmoothing in Hypergraph Neural Networks
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中文总结 AI 辅助
研究超图神经网络过平滑问题，从动力系统视角出发，提出反应扩散框架HNRD，通过定义算子解释消息传递为扩散过程，分析得出过平滑是能量耗散现象，HNRD能补偿耗散稳定变化，实验证明其优于基线，为深度超图架构设计提供框架。
中文摘要 AI 辅助
高阶耦合增强了超图神经网络（HGNNs）的表达能力，但由于强大的多路特征混合，也加剧了深度传播中的表示坍缩。本文从动力系统角度研究超图过平滑问题，开发了用于深度抗性超图学习的反应扩散框架。通过定义超图梯度和散度算子，将消息传递解释为关联级别的扩散过程。对纯扩散的分析表明，其连续半流会指数收缩节点表示的无零模分量并使狄利克雷能量趋于零，揭示了超图过平滑是一种内在的横向能量耗散现象。基于此分析，提出了超图神经反应扩散（HNRD），引入作用于横向分量的反应机制来补偿扩散引起的耗散并稳定判别性变化。建立了所提出动力学的全局适定性，并证明无零模狄利克雷能量保持远离零的有界性。前向欧拉离散化提供了具有深度传播稳定性条件的实用HNRD层。在基准和合成异质超图上的实验表明，HNRD始终优于代表性超图基线。深度、鲁棒性和效率分析进一步表明，HNRD在深度传播和扰动下保持稳定性能和非零狄利克雷能量。这些结果为设计在不发生表示坍缩的情况下保持高阶表达能力的深度超图架构提供了一个有原则的动力框架。
英文摘要
Higher-order couplings enhance the expressive power of hypergraph neural networks (HGNNs), but they also intensify representation collapse in deep propagation due to strong multi-way feature mixing. This work investigates hypergraph oversmoothing from a dynamical-systems perspective and develops a reaction--diffusion framework for depth-resistant hypergraph learning. By defining hypergraph gradient and divergence operators, we interpret message passing as an incidence-level diffusion process. The analysis of pure diffusion shows that its continuous semiflow exponentially contracts the null-mode-free component of node representations and drives the Dirichlet energy to zero, revealing hypergraph oversmoothing as an intrinsic transverse-energy dissipation phenomenon. Motivated by this analysis, we propose Hypergraph Neural Reaction--Diffusion (HNRD), which introduces a reaction mechanism acting on the transverse component to compensate diffusion-induced dissipation and stabilize discriminative variations. We establish global well-posedness of the proposed dynamics and prove that the null-mode-free Dirichlet energy remains bounded away from zero. A forward-Euler discretization provides a practical HNRD layer with a stability condition for deep propagation. Experiments on benchmark and synthetic heterophilic hypergraphs demonstrate that HNRD consistently improves over representative hypergraph baselines. Depth, robustness, and efficiency analyses further show that HNRD preserves stable performance and nonzero Dirichlet energy under deep propagation and perturbations. These results provide a principled dynamical framework for designing deep hypergraph architectures that maintain higher-order expressiveness without representation collapse.
发表机构
- School of Mathematics and Statistics, Shandong University, Weihai（山东大学（威海）数学与统计学院）
- Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）
- University of Chinese Academy of Sciences（中国科学院大学）
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