  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.
