Neural-network preconditioners for solving the Dirac equation in lattice gauge theory

TL;DR

This paper introduces neural-network-based preconditioners to speed up solving the Wilson-Dirac equation in lattice gauge theory, demonstrating improved convergence and scalability across different lattice volumes.

Contribution

The work presents a novel neural-network preconditioning method that accelerates lattice gauge theory computations and can be transferred across different lattice sizes.

Findings

Neural-network preconditioners outperform traditional methods in convergence speed.

Preconditioners trained on small lattices remain effective on larger lattices.

The approach enables scalable solutions for higher-dimensional lattice field theories.

Abstract

This work develops neural-network--based preconditioners to accelerate solution of the Wilson-Dirac normal equation in lattice quantum field theories. The approach is implemented for the two-flavor lattice Schwinger model near the critical point. In this system, neural-network preconditioners are found to accelerate the convergence of the conjugate gradient solver compared with the solution of unpreconditioned systems or those preconditioned with conventional approaches based on even-odd or incomplete Cholesky decompositions, as measured by reductions in the number of iterations and/or complex operations required for convergence. It is also shown that a preconditioner trained on ensembles with small lattice volumes can be used to construct preconditioners for ensembles with many times larger lattice volumes, with minimal degradation of performance. This volume-transferring technique amortizes the training cost and presents a pathway towards scaling such preconditioners to lattice field theory calculations with larger lattice volumes and in four dimensions.