Neural Schur Complement Operators for Coarse-to-Fine Neural Operator Transfer

Neural operators for partial differential equations are advertised as discretization-flexible, yet cross-grid deployment routinely loses accuracy that same-grid validation does not predict. The natural label-free remedy of monitoring a commutator gap fails for a structural reason: any criterion built from $R\,T_f\,P$ is blind to the kernel of the restriction $R$. The mathematically correct learning target is the algebraic Schur complement detail map, and we instantiate it as a Neural Schur Complement Operator that preserves the coarse prediction by construction, targets the energy-optimal Schur correction, admits an additive energy-norm error decomposition, and inherits a projected-CG convergence rate as a polishing step. On hard finite-volume Darcy with six orders of magnitude contrast, the method delivers a $1.67$ to $6.21\times$ improvement over zero-shot lift at $84.2$ to $103.1\%$ oracle-gain recovery, matches direct fine training at one-tenth the fine-label budget, and reduces design regret in a fine-solver-validated well-placement loop using a single fine solve per design.