Neural solutions of coupled ghost and gluon Dyson–Schwinger equations in Landau gauge

This paper investigates the application of physics-informed neural networks (PINNs) to solve the coupled ghost and gluon Dyson-Schwinger equations (DSEs) in four-dimensional Landau-gauge Yang-Mills theory. Unlike traditional methods that rely on direct fixed-point iteration on a discretized grid, this approach uses a neural representation trained exclusively on the renormalized equation residuals. The author compares the neural solution against a conventional fixed-point solver to assess accuracy, stability, and sensitivity to physical parameters.

The neural network successfully reproduces the ghost and gluon propagators, achieving percent-level agreement with direct numerical solutions. The method demonstrates robust stability across different network initializations, architecture sizes, and integration grids. Furthermore, the neural solver correctly captures the MiniMOM ultraviolet running and the sign change of the gluon Schwinger function, which is a key indicator of reflection positivity violation. The study finds that the numerical errors inherent in the neural approach are significantly smaller than the variations caused by different models of the three-gluon vertex, suggesting that the truncation of the equations themselves is the primary source of uncertainty.

Functional equations like DSEs are essential for understanding nonperturbative phenomena in quantum chromodynamics, such as confinement and dynamical chiral symmetry breaking. As these systems become increasingly complex, traditional numerical methods may face limitations in scaling or handling high-dimensional integrals. This work demonstrates that neural representations provide a viable, continuous, and differentiable alternative for solving these integral equations, offering a flexible framework that can be extended to more complex truncations or different gauge theories without requiring pre-existing propagator data.