Self-Supervised Coarse Spaces for Structured Multigrid Failure Modes

Can a learned representation improve a structured multigrid solver after strong classical tuning? We train a Fourier Neural Operator to produce a low-dimensional coarse space using an unsupervised Rayleigh-quotient loss, which consumes neither solution labels nor eigensolves. We evaluate two failure modes. For rotated anisotropic diffusion in 2D, default Ruge-Stueben algebraic multigrid (AMG) takes 106.5 iterations, versus 46.7 for geometric multigrid (GMG); the learned space reduces AMG to 55.4. A stronger classical baseline changes the conclusion: evolution strength-of-connection reduces AMG to 32.6 without learning, and the same learned checkpoint gives a further 18.4% reduction to 26.6. For topologically complex 3D mazes, a held-out, multi-seed study finds a 20.2% mean reduction for GMG3D. Transfer is sharply scoped. Rotated anisotropy is negative in 3D on average, and the maze-trained network at its original rank is harmful on real Bentheimer micro-CT data. Increasing rank and distilling separately on each real operator recovers a 75.5% reduction relative to GMG3D, but still needs 2.1 times the AMG iterations. Paired end-to-end timing with a separate operator-only checkpoint shows no single-solve speedup over evolution AMG; repeated solves only approach break-even. The representation complements classical solver design rather than replacing it.