Neural-Network Holographic Model of the QCD Phase Transition under Lattice and HRG Constraints
Within a neural-network-based holographic framework, we incorporate lattice QCD (LQCD) and Hadron Resonance Gas (HRG) data to train the model and predict the location of the QCD critical endpoint (CEP). The training dataset consists of the entropy density, baryon number susceptibility, and baryon density. The metric warp factor $A(z)$ and the gauge kinetic function $f(z)$ are parameterized by neural networks and determined through the training procedure. The resulting model reproduces the equation of state at vanishing chemical potential in good agreement with both LQCD and HRG data. Extending the analysis to finite chemical potential, we solve the equations of motion and obtain thermodynamic observables consistent with LQCD results at finite density. After incorporating the HRG constraints, the predicted position of the CEP shifts toward larger chemical potentials compared to recent studies. We further employ symbolic regression to derive analytic expressions for $A(z)$ and $f(z)$, providing convenient functional forms for future phenomenological applications. Finally, we perform a data-driven validation using synthetic thermodynamic data generated from an existing analytical holographic model. The neural-network framework reproduces the corresponding CEP location with good accuracy, showing close agreement within numerical uncertainties.