Graph Neural Networks (GNNs) have emerged as one of the leading approaches for machine learning on graph-structured data. Despite their great success, critical computational challenges such as over-smoothing, over-squashing, and limited expressive power continue to impact the performance of GNNs. In this study, inspired from the time-reversal principle commonly utilized in classical and quantum physics, we reverse the time direction of the graph heat equation. The resulted reversing process yields a class of high pass filtering functions that enhance the sharpness of graph node features. Leveraging this concept, we introduce the Multi-Scaled Heat Kernel based GNN (MHKG) by amalgamating diverse filtering functions' effects on node features. To explore more flexible filtering conditions, we further generalize MHKG into a model termed G-MHKG and thoroughly show the roles of each element in controlling over-smoothing, over-squashing and expressive power. Notably, we illustrate that all aforementioned issues can be characterized and analyzed via the properties of the filtering functions, and uncover a trade-off between over-smoothing and over-squashing: enhancing node feature sharpness will make model suffer more from over-squashing, and vice versa. Furthermore, we manipulate the time again to show how G-MHKG can handle both two issues under mild conditions. Our conclusive experiments highlight the effectiveness of proposed models. It surpasses several GNN baseline models in performance across graph datasets characterized by both homophily and heterophily.

翻译：图神经网络已成为处理图结构数据的主流机器学习方法之一。尽管取得了巨大成功，但过平滑、过挤压以及表达能力受限等关键计算难题仍持续影响着图神经网络的性能。本研究受经典物理学与量子物理学中常用的时间反演原理启发，对图热方程的时间方向进行逆转。由此产生的逆过程生成了一类能够增强图节点特征锐利度的高通滤波函数。基于这一概念，我们通过融合不同滤波函数对节点特征的影响，提出了一种基于多尺度热核的图神经网络MHKG。为探索更灵活的滤波条件，我们进一步将MHKG泛化为G-MHKG模型，并详细阐释了各元素在控制过平滑、过挤压及表达能力中所起的作用。值得注意的是，我们证明上述所有问题均可通过滤波函数的性质进行表征与分析，并揭示了过平滑与过挤压之间的权衡关系：增强节点特征锐利度会使模型更易遭受过挤压问题影响，反之亦然。此外，我们通过再次操控时间变量，展示了G-MHKG如何在温和条件下同时应对这两个问题。最终实验验证了所提模型的有效性：在同质性及异质性图数据集上，该模型均超越了多个图神经网络基准模型的性能表现。