Mesh-Admissible State Updates in Graph Neural Simulators via Mesh–GENERIC Compatibility

Graph neural simulators (GNSs) learn state evolution on discretized physical domains, but trajectory accuracy alone does not certify that a learned update is physically admissible . Structure-preserving methods already incorporate physical and geometric structure into graph - and mesh-based simulators . We ask when the GENERIC reversible–dissipative algebraic contract can hold on a prescribed mesh state space, with its metric, admissible directions, and boundary ports governing the executed dynamics . We call this condition Mesh-GENERIC Compatibility (MGC) and give a constructive sufficient condition that enforces skew adjointness, positive semidefiniteness, and energy–entropy degeneracy for every admissible neural readout, independent of training . MAGNS realizes this construction through neural dynamical assembly, which maps learned constitutive and operator quantities into the MGC-compatible operator family before numerical integration. Controlled interventions test integration, boundary exchange, operator assembly, and Hodge-space admissibility. On long-horizon acoustics, MAGNS matches the strongest structured baseline in median rollout accuracy while reducing mean rollout MSE by 6.3% and attaining a smaller common-chart energy gap. Darcy interventions show that changing the declared Hodge space controls harmonic circulation with fixed neural weights, while errors in the remaining flux components are nearly unchanged.