Neural-Network Holographic Model of the QCD Phase Transition under Lattice and HRG Constraints

Abstract

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.

Figures (5)

• Figure 1: Comparison of NNHM predictions with LQCD results for thermodynamic quantities at vanishing chemical potential. The blue solid lines represent the NNHM results, while the red points with error bars denote the (2+1)--flavor LQCD data from Ref. HotQCD:2014kolBellwied:2015lba. For reference, the green solid lines show the HRG model calculations, and the purple dashed lines indicate the SB limits. The plotted quantities are: entropy density $s$, energy density $\epsilon$, pressure $P$, trace anomaly $\epsilon - 3P$, second-order baryon susceptibility $\chi_2^B$, squared speed of sound $C_s^2$, specific heat $C_V$, and free energy $F$, all appropriately scaled by powers of temperature $T$.
• Figure 2: Comparison of NNHM predictions with (2+1)-flavor LQCD results for the thermodynamic EoS at finite chemical potential, the shadow region represents the LQCD results, while the dashed line shows the calculations from NNHM. The panels display the temperature dependence of: (a) scaled baryon number density $\rho/T^3$, (b) second-order baryon susceptibility $\chi_2^B$, (c) scaled pressure $P/T^4$, and (d) scaled energy density $\epsilon/T^4$. The LQCD data are taken from Ref. Bollweg:2022rps (for $\rho$ and $\chi_2^B$) and Ref. Bazavov:2017dus (for $\epsilon$ and $P$).
• Figure 3: QCD phase structure obtained within the data-driven NNHM. (a) Determination of the pseudo-critical temperature from the entropy density, where the inflection point of $s/T^3$ is identified through the tangent construction. (b) The QCD phase diagram in the $T$--$\mu_B$ plane. The crossover region is bounded by two pseudo-critical indicators: the minimum of the squared speed of sound ($C_s^2$, blue dotted line) and the entropy inflection line (green dotted line). These lines become indistinguishable at the CEP, located at $(T, \mu_B)=(0.089, 0.922)\mathrm{GeV}$ (red circle), beyond which a first-order phase transition line (solid black curve) emerges. For comparison, CEP estimates from the $V(\phi)$ EMD model (orange triangle) Yang:2026brr, the merged equation-of-state model including LQCD and HRG constraints (purple square) Yang:2026brr, and a potential-reconstruction EMD model (brown hexagon) are shown Chen:2025goz. The yellow diamond represents a synthetic-data benchmark, in which thermodynamic data generated from a known analytical EMD model are used to train the NNHM, demonstrating that the neural network can reproduce the corresponding CEP location within numerical uncertainties and confirming the data-driven nature of the reconstruction.
• Figure 4: Comparison of the neural-network numerical solutions (gray solid lines) with the analytical expressions obtained from symbolic regression (colored dashed lines). The left panel displays the warp factor $A(z)$, while the right panel shows the gauge kinetic function $f(z)$. The nearly perfect overlap ($R^2 > 0.999$) demonstrates that the derived analytical formulas capture the essential nonlinear dynamics and pole structures of the holographic model.
• Figure 5: Three-dimensional distributions of thermodynamic observables obtained from the reconstructed neural-network holographic model on the $(T,\mu_B)$ plane. The panels show, from left to right and top to bottom, the entropy density $s/T^3$, the second-order baryon number susceptibility $\chi_2^B$, the energy density $\epsilon/T^4$, the pressure $p/T^4$, the trace anomaly $(\epsilon-3p)/T^4$, the squared speed of sound $C_s^2$, the specific heat $C_V/T^3$, and the baryon number density $\rho/T^3$. These surface plots illustrate the global thermodynamic structure of the model at finite temperature and baryon chemical potential, and provide a direct visualization of the nontrivial density dependence.