Residual-Conditioned Subspace Correction for Helmholtz Neural Operators

Helmholtz discretizations can share Poisson’s sparsity pattern while exhibiting indefiniteness and near-resonant sensitivity, motivating neural-operator designs informed by matrix properties. We propose residual-conditioned subspace correction (ReSCo), which combines a pretrained multigrid neural operator with a fixed POD error basis and a learned coefficient-recovery module. The module uses physical residuals and projected features to predict global wavefield corrections. To improve generalization, we train it on samples excluded from backbone training. On the Caltech Helmholtz validation benchmark across three random seeds, the method reduces mean relative $L^2$ error from 0.00998 to 0.00619, achieving a mean per-seed improvement of 37.27%, compared with 6.2% when correction training uses backbone-seen samples. These results demonstrate the value of combining matrix-informed correction with training on residuals.