Gauge-equivariant multigrid neural networks

In Lattice QCD, the solution of the Dirac equation often dominates the wall-clock time of the simulations. This time can be greatly reduced if we can find a suitable preconditioner. We apply machine-learning methods to this problem. In particular, we show how multigrid preconditioners for the Wilson-clover Dirac operator 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.