Hybrid Residual Correction of VMC Charmonium Masses with a Screened Funnel Interaction
In this study, we combine residual correction with the physical treatment of charmonium masses within a Quantum Chromodynamics (QCD) motivated potential-model framework via variational Monte Carlo (VMC). The aim is not to propose a new charmonium spectrum, since this sector has already been examined extensively through different potential models. Instead, the main objective is to evaluate how effectively a Machine Learning (ML) correction can improve a VMC baseline when both stages are built on the same screened funnel potential. In this workflow, the screened interaction provides the physical input and determines the underlying mass structure. The proposed run uses the variational method via VMC and deterministic eigenvalue diagonalization in the process of mass calculation. In total, ten charmonium states are calculated, among which seven use the experimental reference masses. The VMC step generates a large set of configurations and local energy estimators that are fed into the neural network (NN) residual corrector at the sample level, while the corrections and experimental uncertainties are collected at the state level. It then learns the residual difference between the raw physics baseline and the internal reference targets. This hybrid procedure reduces the systematic mass offset of the raw calculation across the studied states. For the experimentally verified seven states, the correction reduces the MAE from $438.1~\mathrm{MeV}$ to $24.1~\mathrm{MeV}$, corresponding to a $94.5\%$ reduction. These results show that ML can serve as a residual-correction layer for potential-model spectroscopy.