  Recent works have shown that physics-inspired architectures allow the
training of deep graph neural networks (GNNs) without oversmoothing. The role
of these physics is unclear, however, with successful examples of both
reversible (e.g., Hamiltonian) and irreversible (e.g., diffusion) phenomena
producing comparable results despite diametrically opposed mechanisms, and
further complications arising due to empirical departures from mathematical
theory. This work presents a series of novel GNN architectures based upon
structure-preserving bracket-based dynamical systems, which are provably
guaranteed to either conserve energy or generate positive dissipation with
increasing depth. It is shown that the theoretically principled framework
employed here allows for inherently explainable constructions, which
contextualize departures from theory in current architectures and better
elucidate the roles of reversibility and irreversibility in network
performance.
