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-St\"uben 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. These results position the representation as a complement to classical solver design, not a replacement for it.