A novel gauge-equivariant neural-network architecture for preconditioners in lattice QCD

TL;DR

This work introduces a novel gauge-equivariant neural-network architecture for preconditioning the Dirac equation in the regime where critical slowing down occurs and studies the behavior of this preconditioner as a function of topological charge and lattice volume to show it mitigates critical slowing down.

Abstract

Lattice QCD simulations are computationally expensive, with the solution of the Dirac equation being the major computational bottleneck of many calculations. We introduce a novel gauge-equivariant neural-network architecture for preconditioning the Dirac equation in the regime where critical slowing down occurs. We study the behavior of this preconditioner as a function of topological charge and lattice volume and show that it mitigates critical slowing down. We also show that this preconditioner transfers to unseen gauge configurations without any retraining, therefore enabling applications not possible with competing methods.

Figures (5)

• Figure 1: Exemplary network architecture using linear layers (L) and parallel-transport layers (PT). For deeper networks, the highlighted block consisting of a PT and an L layer can be repeated, with potentially different parallel-transport paths in each of the PT layers.
• Figure 2: Left: Evolution of the residual during GMRES solves with and without preconditioners, comparing the choices $P_s$ and $P_\ell$ of parallel-transport paths. Networks are trained with the cost function $C_N$ with $N=10$ filter iterations. Right: Operator applications needed in GMRES solves to reach a residual of $10^{-18}$ with and without preconditioners, where all neural networks use the set $P_\ell$ of parallel-transport paths. Both plots show data for a lattice size of $8^3\times 16$, a quenched gauge configuration at $\beta=6$ with topological charge $Q=1$, and bare mass parameter $m=-0.555$ (such that $m-m_\textrm{crit}\approx5\times10^{-4}$).
• Figure 3: Operator applications needed in GMRES solves to reach a residual of $10^{-18}$ with and without preconditioners, depending on the bare mass parameter. An $8^3\times16$ lattice is used, and the topological charge of the gauge configuration is $Q=0$ (left) and $Q=1$ (right). Models are trained individually for each bare mass parameter. The dashed vertical line denotes the critical mass, defined as the largest bare mass parameter for which an eigenvalue of $D$ has zero real part.
• Figure 4: Same as \ref{['fig:CSD_8c16']}, but for a $16^3\times32$ lattice and topological charges $Q=0$ (left) and $Q=4$ (right).
• Figure 5: Application of a model trained on the $8^3\times16$ lattice with a gauge configuration with $Q=0$ and $m=-0.56$ to a $8^3\times16$ lattice with a gauge configuration with $Q=1$ and different masses (left), and to a $16^3\times32$ lattice with a gauge configuration with $Q=0$ and different masses (right).