Physics-informed neural network approach to quark-antiquark color flux tube

We introduce a physics-informed neural network (PINNs) framework for modeling the spatial distribution of chromodynamic fields induced by quark-antiquark pairs, based on lattice Monte Carlo simulations. In contrast to conventional neural networks, PINNs incorporate physical laws—expressed here as differential equations governing type-II superconductivity—directly into the training objective. By embedding these equations into the loss function, we guide the network to learn physically consistent solutions. Adopting an inverse problem approach, we extract the parameters of the superconducting equations from lattice QCD data and subsequently solve them. To accommodate physical boundary conditions, we recast the system into an integrodifferential form and extend the analysis within the fractional PINNs framework. The accuracy of the reconstructed field distribution is assessed via relative <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:msub><a:mi>L</a:mi><a:mn>2</a:mn></a:msub></a:math>-error norms. We further extract physical observables such as the string tension and the mean width of the flux tube, offering quantitative insight into the confinement mechanism. This method enables the reconstruction of color field profiles as functions of quark-antiquark separation without recourse to predefined parametric models. Our results illuminate aspects of the dual Meissner effect and highlight the promise of data-driven strategies in addressing nonperturbative challenges in quantum chromodynamics.