AI for Pattern Hunter: Application in Wilson Loop of 2D Lattice Yang-Mills Theory

TL;DR

This work demonstrates that a Transformer can learn a direct mapping from the shape of Wilson loops in 2D lattice Yang–Mills theory to their vacuum expectation values, revealing a meaningful link between loop geometry and analytic results. By tokenizing Wilson-loop shapes and encoding the polynomial dependence on $u=igra rac{1}{N} ext{Tr}(U_P) igra$ and the coupling $oldsymbol{}$, the model achieves high accuracy across loop lengths up to 16 and across varied hyperparameters. The study also shows how training data size and mixed-length training influence learning efficiency and generalization, highlighting threshold effects and the potential for accelerated pattern discovery to inspire rigorous derivations. Overall, this pattern-hunting approach provides a foundation for extending ML-assisted analytical insights to higher-dimensional lattice gauge theories and for guiding future physics-inspired model design.

Abstract

We employ the Transformer to learn patterns in two-dimensional lattice Yang-Mills theory. Specifically, we represent both Wilson loops and their expectation values as tokenized sequences. Taking the shape of Wilson loops as input, the model successfully predicts expectation values with high accuracy, indicating a meaningful connection between loop geometry and physical results. Our study differs from prior machine learning applications in lattice QCD by emphasizing analytical structures rather than numerical computations. We explore model performance under varying hyperparameters, training data sizes, and sequence lengths. This work serves as a first step toward extending such methods to higher dimensions and inspiring rigorous analytical derivations.

Figures (11)

• Figure 1: A Wilson loop of length 8.
• Figure 2: The learning curves for different cases.
• Figure 3: The impact of the dimension. Each diagram has the same layer and head. In each diagram, there are two learning curves corresponding to the dimensions $128$ and $256$, respectively.
• Figure 7: Learning curves for various proportions of training dataset and different initial condition of models. For models in each sub picture, we randomly generate 5 different initial parameters and repeat the training.
• Figure 8: Learning curves for various proportions of training dataset. Each line represents the mean value of five experiments with different initial condition.