Graph neural networks (GNNs) are widely used in domains like social networks and biological systems. However, the locality assumption of GNNs, which limits information exchange to neighboring nodes, hampers their ability to capture long-range dependencies and global patterns in graphs. To address this, we propose a new inductive bias based on variational analysis, drawing inspiration from the Brachistochrone problem. Our framework establishes a mapping between discrete GNN models and continuous diffusion functionals. This enables the design of application-specific objective functions in the continuous domain and the construction of discrete deep models with mathematical guarantees. To tackle over-smoothing in GNNs, we analyze the existing layer-by-layer graph embedding models and identify that they are equivalent to l2-norm integral functionals of graph gradients, which cause over-smoothing. Similar to edge-preserving filters in image denoising, we introduce total variation (TV) to align the graph diffusion pattern with global community topologies. Additionally, we devise a selective mechanism to address the trade-off between model depth and over-smoothing, which can be easily integrated into existing GNNs. Furthermore, we propose a novel generative adversarial network (GAN) that predicts spreading flows in graphs through a neural transport equation. To mitigate vanishing flows, we customize the objective function to minimize transportation within each community while maximizing inter-community flows. Our GNN models achieve state-of-the-art (SOTA) performance on popular graph learning benchmarks such as Cora, Citeseer, and Pubmed.

翻译：图神经网络（GNN）广泛应用于社交网络和生物系统等领域。然而，GNN 的局部性假设将信息交换限制在相邻节点之间，这削弱了其捕捉图中长程依赖关系和全局模式的能力。为解决此问题，受最速降线问题启发，我们提出一种基于变分分析的新归纳偏置。我们的框架建立了离散 GNN 模型与连续扩散泛函之间的映射关系，从而能够在连续域中设计特定于应用的目标函数，并构建具有数学保证的离散深层模型。为解决 GNN 中的过度平滑问题，我们分析了现有的逐层图嵌入模型，发现其等价于图梯度的 l2 范数积分泛函，而这正是导致过度平滑的原因。类似于图像去噪中的边缘保持滤波器，我们引入全变分（TV）以将图扩散模式与全局社区拓扑对齐。此外，我们设计了一种选择性机制，以应对模型深度与过度平滑之间的权衡，该机制可轻松集成到现有 GNN 中。进一步，我们提出一种新颖的生成对抗网络（GAN），通过神经传输方程预测图中的传播流。为缓解流消失问题，我们定制目标函数以最小化社区内部传输，同时最大化社区间流量。我们的 GNN 模型在流行的图学习基准（如 Cora、Citeseer 和 Pubmed）上达到了最先进的（SOTA）性能。