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We demonstrate that gauge-equivariant pooling and unpooling layers canperform as well as traditional restriction and prolongation layers in multigridpreconditioner models for lattice QCD. These layers introduce a gauge degree offreedom on the coarse grid, allowing for the use of explicitlygauge-equivariant layers on the coarse grid. We investigate the construction ofcoarse-grid gauge fields and study their efficiency in the preconditionermodel. We show that a combined multigrid neural network using a Galerkinconstruction for the coarse-grid gauge field eliminates critical slowing down.

Gauge-equivariant pooling layers for preconditioners in lattice QCD