Learning What Multigrid Leaves Behind: Objective Alignment and Its Limits

A learned coarse space adds a low-rank correction to a multigrid preconditioner, but which space should the network predict? We separate three requirements. Target: the training target should complement the deployed base preconditioner, not only the operator's low-energy modes. Prediction: a network must predict that target on new operators. Cost: the correction must beat strong classical solvers in total time. For a standard two-level balancing correction, a known result says the condition-number-optimal space consists of the directions the base preconditioner reduces most slowly. We train an FNO on a finite solver target that approximates these directions by repeatedly applying the preconditioner's error propagator to random vectors. With architecture, rank, training operators, and deployment fixed, this target raises the mean iteration reduction on 24 held-out 2D operators from 17.6% (operator-only Rayleigh training) to 31.7%, beating even the Rayleigh objective's exact optimum (18.8%), and targets crossed between two preconditioners lose on every operator. The target requirement therefore holds in this 2D family, and a useful target also exists in a 3D obstacle family. Prediction holds only inside the training band, where it is slightly better than reusing a stored target; it fails on disjoint angles, on lognormal coefficients, and in 3D. Cost favors classical solvers: the corrected preconditioner needs about as many iterations as two V-cycles and more flops, and BoomerAMG is faster in total time.