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

Unifying over-smoothing and over-squashing in graph neural networks: A physics informed approach and beyond