Operator-Consistent Graph Neural Networks for Learning Diffusion Dynamics on Irregular Meshes

Learning partial differential equation dynamics on irregular meshes requires preserving the geometric and algebraic structure of the underlying differential operators. In such settings, classical discretizations are difficult to apply and analytical solutions are often unavailable, limiting the applicability of supervision-based learning. We propose an operator-consistent graph neural network with physics-informed constraints for modeling PDE evolution under these conditions. The approach represents the spatial domain as a graph and couples node and edge dynamics through a consistency loss induced by the graph incidence structure, enforcing the discrete gradient divergence relationship during temporal rollout. By preserving operator-level structure, the model improves stability when learning without explicit solution supervision. Experiments on diffusion processes over evolving meshes and real-world scanned surfaces demonstrate improved temporal stability and accuracy compared with graph convolutional and multilayer perceptron baselines, approaching the performance of classical implicit solvers.