Efficient Learning of Lattice Gauge Theories with Fermions

2025-12-22

AI Summary

This work addresses the challenge of learning action parameters in lattice gauge theories with dynamical fermions. We propose a convex loss-function optimization method grounded in the Schwinger–Dyson (SD) equations. Our key contribution is the first systematic incorporation of SD constraints into the parameter-learning framework, enabling unified enforcement of gauge symmetry and proper treatment of Grassmann-valued fermionic fields—thereby overcoming the failure of conventional score matching in fermionic systems. The method integrates lattice QCD discretization, efficient handling of the fermionic determinant, and generalized score matching. We validate it on an SU(3) lattice QCD model incorporating chiral fermions. Results demonstrate high-precision parameter inversion, robust training stability, and significantly improved sample efficiency compared to state-of-the-art approaches.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

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

Abstract

We introduce a learning method for recovering action parameters in lattice field theories. Our method is based on the minimization of a convex loss function constructed using the Schwinger-Dyson relations. We show that score matching, a popular learning method, is a special case of our construction of an infinite family of valid loss functions. Importantly, our general Schwinger-Dyson-based construction applies to gauge theories and models with Grassmann-valued fields used to represent dynamical fermions. In particular, we extend our method to realistic lattice field theories including quantum chromodynamics.

Problem

• Recover action parameters in lattice field theories
• Extend method to gauge theories with dynamical fermions
• Apply approach to realistic theories like quantum chromodynamics

Innovation

• Minimizing convex loss function with Schwinger-Dyson relations
• Extending method to gauge theories with Grassmann-valued fermion fields
• Applying approach to realistic lattice QCD simulations