Design Your Own Universe: A Physics-Informed Agnostic Method for Enhancing Graph Neural Networks

Physics-informed Graph Neural Networks have achieved remarkable performance in learning through graph-structured data by mitigating common GNN challenges such as over-smoothing, over-squashing, and heterophily adaption. Despite these advancements, the development of a simple yet effective paradigm that appropriately integrates previous methods for handling all these challenges is still underway. In this paper, we draw an analogy between the propagation of GNNs and particle systems in physics, proposing a model-agnostic enhancement framework. This framework enriches the graph structure by introducing additional nodes and rewiring connections with both positive and negative weights, guided by node labeling information. We theoretically verify that GNNs enhanced through our approach can effectively circumvent the over-smoothing issue and exhibit robustness against over-squashing. Moreover, we conduct a spectral analysis on the rewired graph to demonstrate that the corresponding GNNs can fit both homophilic and heterophilic graphs. Empirical validations on benchmarks for homophilic, heterophilic graphs, and long-term graph datasets show that GNNs enhanced by our method significantly outperform their original counterparts.