Learning Lattice Variables for Matrix-Free Wilson-Dirac Preconditioning in Lattice QCD

Solving large, sparse, and ill-conditioned linear systems dominates the computational cost of lattice quantum chromodynamics (LQCD). Preconditioners effectively accelerate the convergence of iterative solvers but generally need careful design and tuning. This work develops an encoder--representation--decoder framework to produce a matrix-free neural preconditioner for the Wilson-Dirac operator. The encoder maps the gauge configuration $U$ to a representation $V$, which the decoder turns into a Hermitian positive-definite (HPD) preconditioner $M^{-1}$ for the Krylov solver. Because the representation consists of matrix-valued lattice fields, the decoder is assembled from lattice operators and never forms the actual preconditioning matrix---a computationally prohibitive operation for typical systems. No learned quantity in either the encoder or the decoder is tied to the lattice volume, so a trained encoder--decoder pair applies unchanged at any lattice size. Across four encoders and four decoders, we train the framework using gauge configurations of lattice size $L^4=8^4$ against the unrolled preconditioned Conjugate Gradient (PCG) objective and show that the generated preconditioners effectively reduce the number of iterations required for convergence, achieving a 7.8$\times$ reduction compared to the unpreconditioned case. In addition, we directly apply the trained encoder--decoder pairs to a larger system of $L^4=16^4$, where the convergence behavior and relative ordering are preserved ($\sim$8$\times$ iteration reduction) without retraining.