Gauge-equivariant multigrid neural networks

In lattice QCD simulations, the most time-consuming element is typically the solution of the Dirac equation in the presence of a given gauge field. The current state of the art is to use a multigrid preconditioner to reduce the condition number of the Dirac operator matrix. We show how such preconditioners can be constructed using gauge-equivariant neural networks. For the multigrid solve we employ parallel-transport convolution layers. For the multigrid setup we consider two versions: the standard construction based on the near-null space of the operator, and a gauge-equivariant construction using pooling and subsampling layers. We show that both versions eliminate critical slowing down.