While Hyperbolic Graph Neural Network (HGNN) has recently emerged as a powerful tool dealing with hierarchical graph data, the limitations of scalability and efficiency hinder itself from generalizing to deep models. In this paper, by envisioning depth as a continuous-time embedding evolution, we decouple the HGNN and reframe the information propagation as a partial differential equation, letting node-wise attention undertake the role of diffusivity within the Hyperbolic Neural PDE (HPDE). By introducing theoretical principles \textit{e.g.,} field and flow, gradient, divergence, and diffusivity on a non-Euclidean manifold for HPDE integration, we discuss both implicit and explicit discretization schemes to formulate numerical HPDE solvers. Further, we propose the Hyperbolic Graph Diffusion Equation (HGDE) -- a flexible vector flow function that can be integrated to obtain expressive hyperbolic node embeddings. By analyzing potential energy decay of embeddings, we demonstrate that HGDE is capable of modeling both low- and high-order proximity with the benefit of local-global diffusivity functions. Experiments on node classification and link prediction and image-text classification tasks verify the superiority of the proposed method, which consistently outperforms various competitive models by a significant margin.

通过将深度视为连续时间嵌入演化的方法，我们在本文中解耦了超椭圆图神经网络 (HGNN) 并将信息传播重新构造为偏微分方程 (Partial Differential Equation, PDE)，让节点注意力承担超椭圆神经 PDE(Hyperbolic Neural PDE, HPDE)中的传导性作用。通过引入理论原则，如非欧几里得流形上的场和流、梯度、散度和传导性，我们讨论了用于形成数值 HPDE 解算器的隐式和显式离散化方案。进一步，我们提出了超椭圆图扩散方程(Hyperbolic Graph Diffusion Equation, HGDE) - 这是一个灵活的矢量流函数，它可以集成以获得表达力强的超椭圆节点嵌入。通过分析嵌入的潜在能量衰减，我们证明了 HGDE 能够模拟低阶和高阶近邻关系，并具有局部-全局传导函数的好处。节点分类和链路预测以及图像-文本分类任务的实验证明了所提方法的优越性，其性能始终显著优于各种竞争模型。

连续几何感知图扩散：超几何神经偏微分方程