Universality of Gluon Saturation from Physics-Informed Neural Networks

Abstract: The universality of the color dipole amplitude is a cornerstone of high-energy Quantum Chromodynamics (QCD). However, standard phenomenological approaches typically rely on rigid parametric ansatzes and often require ad-hoc geometric adjustments to reconcile inclusive and diffractive measurements. To resolve this tension, we introduce Physics-Informed Neural Networks (PINNs) employing a Teacher--Student'' strategy. The rigorous momentum-space Balitsky-Kovchegov evolution dynamics act as theTeacher,'' constraining the solution manifold, while the network ``Student'' is refined against inclusive HERA $F_2$ data. This approach extracts a model-independent dipole amplitude without assuming initial states. Strikingly, we demonstrate that this amplitude -- without parameter retuning or geometric rescaling -- successfully predicts exclusive $J/ψ$ photoproduction cross-sections. This zero-parameter prediction rigorously confirms the universality of the gluon saturation scale and establishes PINNs as a transformative paradigm for uncovering non-perturbative QCD structures.