SUMNO: Spatially Unrolled Multigrid Neural Operator for Fluid Flow Surrogates

Data-driven surrogate modeling has emerged as a powerful paradigm for accelerating computational fluid dynamics, and hierarchical U-shaped networks, which resemble the structure of classical Geometric Multigrid (GMG) methods, have become prevalent designs. Modern surrogates built on these designs achieve competitive accuracy at a fraction of the cost of numerical solvers; however, accurate prediction of complex flows remains challenging, and no existing architecture leads across them. In such networks, the multigrid analogy mainly concerns the multi-resolution topology, while their components largely follow common vision practice. To address this, this study introduces SUMNO, a multigrid neural operator for fluid flow fields on regular grids that spatially unrolls a V-cycle into a feed-forward U-shaped network. Parameter-free down- and upsampling stand in for restriction and prolongation, and additive skip connections for coarse-grid correction, with normalization-free transfer and skip pathways. Within this macro-architecture, convolutions are used at the fine and intermediate resolutions and shifted-window attention at the coarsest, whose window partitioning is generalized to handle periodicity independently per axis. Evaluated on unseen test configurations across four unsteady and steady flow benchmarks against five representative baselines, SUMNO is best in 5 of 7 settings and ranks second in the remaining two. It reduces the 10-step rollout MSE by about 30% relative to the strongest baseline, and outperforms three baselines with over 130 million parameters using 9.4 million parameters.