Learning to decompose PDE domains with geometry-aware local operators and global communication

Neural operators provide an efficient approach to repeated partial differential equation (PDE) solving, but practical use on complex geometries and large meshes remains constrained by the challenge of balancing local spatial resolution, global physical coupling, and computational cost. A Geometry-Aware Domain-Decomposition Neural Operator (GDDNO) is introduced to organize local computation and global communication through domain decomposition. Within each subdomain, local interactions are encoded by a shared graph operator. The global information is exchanged between the subdomain tokens through attention, and the resulting context is combined with retained cell features for prediction. Overlap domain assembly and flux consistency constraints coordinate predictions between subdomains. GDDNO is evaluated on five RealPDEBench systems, AirfRANS, and a three-dimensional reacting flow dataset. It obtains the lowest simulation-trained relative $L_2$ error on all five RealPDEBench systems. On full-field AirfRANS, the relative $L_2$ error is 0.0815, compared with 0.3511 for the strongest baseline, and the prediction of the aerodynamic coefficients is also the best. On cavity combustion flow, the relative $L_2$ errors for the predicted fields and combustion efficiency are 0.4148 and 0.2112, respectively, both the lowest among the compared models. These results indicate that domain decomposition can be used as an effective strategy to combine local geometric detail with long-range interactions in neural PDE operators.