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：图神经网络已成为图结构数据机器学习中的主流方法之一。尽管取得了巨大成功，但过平滑、过挤压以及表达能力有限等关键计算挑战仍持续影响其性能。本研究受经典物理和量子物理中常用的时间反演原理启发，反向推导了图热方程的时间方向。该反向化过程产生了一类能够增强图节点特征锐度的高通滤波函数。基于此概念，我们通过融合不同滤波函数对节点特征的影响，提出了基于多尺度热核的图神经网络。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，并深入阐释了各元素在控制过平滑、过挤压及表达能力中的作用。值得注意的是，我们证明所有上述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：增强节点特征锐度会加剧过挤压问题，反之亦然。此外，我们通过再次操控时间维度，展示了G-MHKG如何在温和条件下同时处理这两个问题。最终实验验证了所提模型的有效性，在兼具同配性与异配性的图数据集上，其性能超越了多个图神经网络基线模型。