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

2025-09-12

AI Summary

In lattice QCD, solving the Dirac operator’s linear system—Hermitian positive definite, sparse, and highly ill-conditioned—has long suffered from slow convergence of iterative solvers (e.g., conjugate gradient, CG) and high computational overhead in constructing effective preconditioners. This work proposes a **matrix-free neural preconditioner framework**, leveraging deep operator learning to implicitly model efficient linear mappings and integrating the learned preconditioner end-to-end into the CG algorithm. The method enables matrix-free construction and zero-shot cross-scale generalization—i.e., direct deployment across different lattice volumes and gauge configurations without retraining. Validated on the 1+1-dimensional U(1) lattice gauge model, it reduces the system’s condition number significantly and cuts CG iterations by approximately 50%. The approach demonstrates strong multi-scale adaptability, numerical stability, and computational efficiency, offering a scalable alternative to conventional preconditioning strategies in lattice field theory simulations.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding
• Computer Vision: Learning & Optimization for CV

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs

Abstract

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.

Problem

• Accelerating iterative solvers for ill-conditioned Dirac operator systems
• Reducing computational cost of lattice QCD simulations via neural preconditioners
• Enabling zero-shot learning for preconditioners across different lattice sizes

Innovation

• Matrix-free neural preconditioner framework
• Operator learning for linear system acceleration
• Zero-shot learning across lattice sizes