  Neural message passing is a basic feature extraction unit for
graph-structured data considering neighboring node features in network
propagation from one layer to the next. We model such process by an interacting
particle system with attractive and repulsive forces and the Allen-Cahn force
arising in the modeling of phase transition. The dynamics of the system is a
reaction-diffusion process which can separate particles without blowing up.
This induces an Allen-Cahn message passing (ACMP) for graph neural networks
where the numerical iteration for the particle system solution constitutes the
message passing propagation. ACMP which has a simple implementation with a
neural ODE solver can propel the network depth up to one hundred of layers with
theoretically proven strictly positive lower bound of the Dirichlet energy. It
thus provides a deep model of GNNs circumventing the common GNN problem of
oversmoothing. GNNs with ACMP achieve state of the art performance for
real-world node classification tasks on both homophilic and heterophilic
datasets.
