# Matrix-free Neural Preconditioner for the Dirac Operator in Lattice Gauge Theory

arXiv: https://arxiv.org/abs/2509.10378 · HTML: https://mlsanity.com/paper/2509.10378/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 69 references that are in the map.
- First posted on arXiv 12 Sep 2025.

## Builds on (most cited first, 20 of 20)

- [Fully Convolutional Networks for Semantic Segmentation](https://mlsanity.com/paper/1411.4038/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [A physics-informed operator regression framework for extracting data-driven continuum models](https://mlsanity.com/paper/2009.11992/) (2020)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [Deep Learning of Preconditioners for Conjugate Gradient Solvers in Urban Water Related Problems](https://mlsanity.com/paper/1906.06925/) (2019)
- [Neural incomplete factorization: learning preconditioners for the conjugate gradient method](https://mlsanity.com/paper/2305.16368/) (2023)
- [Graph Neural Preconditioners for Iterative Solutions of Sparse Linear Systems](https://mlsanity.com/paper/2406.00809/) (2024)
- [DeepGraphONet: A Deep Graph Operator Network to Learn and Zero-shot Transfer the Dynamic Response of Networked Systems](https://mlsanity.com/paper/2209.10622/) (2022)
- [Machine learning and domain decomposition methods -- a survey](https://mlsanity.com/paper/2312.14050/) (2023)
- [Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows](https://mlsanity.com/paper/2010.02866/) (2020)
- [Gauge-equivariant neural networks as preconditioners in lattice QCD](https://mlsanity.com/paper/2302.05419/) (2023)
- [Neural-network preconditioners for solving the Dirac equation in lattice gauge theory](https://mlsanity.com/paper/2208.02728/) (2022)
- [Gauge-equivariant pooling layers for preconditioners in lattice QCD](https://mlsanity.com/paper/2304.10438/) (2023)
- [Coarsest-level improvements in multigrid for lattice QCD on large-scale computers](https://mlsanity.com/paper/2205.09104/) (2022)
- [MRHS multigrid solver for Wilson-clover fermions](https://mlsanity.com/paper/2211.13719/) (2022)
- [Multiple right hand side multigrid for domain wall fermions with a multigrid preconditioned block conjugate gradient algorithm](https://mlsanity.com/paper/2409.03904/) (2024)

## Abstract (arXiv)

Linear systems arise in generating samples and in calculating observables in lattice quantum chromodynamics~(QCD). Solving the Hermitian positive definite systems, which are sparse but ill-conditioned, involves using iterative methods, such as Conjugate Gradient (CG), which are time-consuming and computationally expensive. Preconditioners can effectively accelerate this process, with the state-of-the-art being multigrid preconditioners. However, constructing useful preconditioners can be challenging, adding additional computational overhead, especially in large linear systems. We propose a framework, leveraging operator learning techniques, to construct linear maps as effective preconditioners. The method in this work does not rely on explicit matrices from either the original linear systems or the produced preconditioners, allowing efficient model training and application in the CG solver. In the context of the Schwinger model U(1) gauge theory in 1+1 spacetime dimensions with two degenerate-mass fermions), this preconditioning scheme effectively decreases the condition number of the linear systems and approximately halves the number of iterations required for convergence in relevant parameter ranges. We further demonstrate the framework learns a general mapping dependent on the lattice structure which leads to zero-shot learning ability for the Dirac operators constructed from gauge field configurations of different sizes.