SUMNO unrolls a multigrid V-cycle into a feed-forward network: 9.4M parameters, best in five of seven flow settings
Synopsis
Fluid surrogate models have long reused U-shaped multiscale designs from vision, but their resampling, skip fusion and normalization were not built for flow fields. SUMNO spatially unrolls a multigrid V-cycle into a feed-forward U-shaped network, using parameter-free downsampling and bilinear upsampling for restriction and prolongation, and shifted-window attention at the coarsest level, with window masks set per-axis periodicity, for global coupling. Across seven settings on four flow benchmarks it is best in five and second in two, cutting 10-step rollout MSE by about 30% on Pinball and RBC, with 9.4M parameters.
Interpretation
Fluid surrogates now have a U-shaped network whose components follow a multigrid cycle rather than vision conventions, and it leads on complex flows. In earlier U-shaped surrogates the multigrid analogy concerned mainly the multi-resolution topology, while resampling, skip fusion and normalization largely followed vision practice. Evaluated on four flow benchmarks across seven settings against five representative baselines, SUMNO is best in five settings and second in two; on Pinball and RBC it reduces 10-step rollout MSE by about 30% relative to the strongest baseline.
The network unrolls one multigrid V-cycle in space: a descending encoder, a coarsest-level solve, and an ascending decoder, matching multigrid smoothing, restriction, coarse-grid correction and prolongation. Restriction and prolongation are not learned operators but parameter-free BlurPool downsampling and bilinear upsampling; skips are fused additively with learnable relaxation factors, and normalization is dropped from transfer and skip pathways. Convolutions at fine and intermediate levels preserve local continuity, shifted-window attention at the coarsest level provides global coupling, and window masks are generated per-axis from periodicity so doubly periodic, cylindrical and non-periodic domains share one rule.
Perspective
The demonstrated scope is two-dimensional incompressible flow on regular grids: three unsteady systems and one steady boundary-value problem, with training, validation and test splits disjoint at the trajectory level (geometry level for LDC). The authors propose extending to irregular or unstructured meshes and three-dimensional flows, and enforcing exact physical invariants such as mass or energy conservation.
The authors note that predicted fields still deviate visibly from ground truth over extended rollouts, for example predicted vortices lag their true positions in the cylinder wake; SUMNO targets regular grids and, like data-driven surrogates generally, provides no formal accuracy or stability guarantees.