A black-box, multilevel algebraic preconditioning framework for conforming finite elements

What the math gives to ML

The paper's transferable asset is a black-box multilevel preconditioner built from a local Gram decomposition rather than from geometric mesh information. Each row of a factor G defines a rank-one SPSD energy, making locality discoverable from row supports; Galerkin coarsening preserves the same Gram form exactly. This suggests a practical second-order optimizer for least-squares neural networks: construct a Jacobian Gram approximation, smooth parameter errors with overlapping local solves, and remove globally coupled low-energy directions through a learned coarse correction. The strongest initial target is damped Gauss–Newton or Levenberg–Marquardt training for small-to-medium regression, PINN, or implicit-network models.