HoloNet: Toward a Unified Einstein-Maxwell-Dilaton Framework of QCD

TL;DR

HoloNet introduces a fully data-driven holographic QCD framework that learns the bulk warp factor $A(z)$ and gauge–dilaton coupling $f(z)$ directly from 2+1 flavor LQCD data at zero chemical potential and embeds these into the five-dimensional Einstein–Maxwell–dilaton theory to reproduce the equation of state and baryon-number fluctuations. By leveraging neural networks along the holographic direction, the approach eliminates assumed functional forms and yields a consistent reconstruction of the potential $V(φ)$ and coupling $f(φ)$, agreeing with holographic renormalization. The model extends to finite density to map the QCD phase diagram and estimate the CEP, reporting a CEP at $(T,μ)=(106\ \mathrm{MeV},730\ \mathrm{MeV})$, while highlighting uncertainties due to extrapolation near the data edge. The work demonstrates that potential-reconstruction and holographic renormalization yield compatible results within the EMD framework and offers a data-driven path to reducing model arbitrariness in holographic QCD.

Abstract

We propose HoloNet, a neural-network framework that unifies lattice QCD(LQCD) thermodynamics and holographic Einstein-Maxwell-Dilaton (EMD) theory within a data-to-holography pipeline. Instead of assuming specific functional forms, HoloNet learns the metric profile $A(z)$ and the gauge-dilaton coupling $f(z)$ directly from 2+1-flavor LQCD data at $μ=0$. These learned functions are embedded into the EMD equations, enabling the model to reproduce the lattice equation of state and baryon number fluctuations with high fidelity. Once trained, HoloNet provides a fully data-driven holographic description of QCD that extends naturally to finite density, allowing us to map the phase diagram and estimate the location of the critical end point (CEP). The reconstructed potential $V(φ)$ and coupling $f(φ)$ agree quantitatively with those obtained from holographic renormalization, demonstrating that HoloNet can consistently bridge different holographic models.

Figures (8)

• Figure 1: HoloNet. A neural network is constructed along the holographic direction. Different bulk spacetimes are placed on a common holographic coordinate and differ only by their horizon locations $z_H$. The functions $A(z)$ and $f(z)$ are implemented as sub-networks (schematically shown as three-layer networks). Hollow nodes represent learnable parameters, while solid nodes represent the fixed equation of motion. The sub-networks output $A(z)$ and $f(z)$ at each layer and feed them into the fixed network to compute thermodynamic quantities.
• Figure 2: The graphs of the $A$ and $f$ functions are shown, with red representing the optimized HoloNet results, black corresponding to an analytically guessed solution, blue depicting the machine learning results obtained from a prescribed functional form Chen:2024ckb, and green indicating the outcomes of Bayesian analysis zhu2025bayesian. Additionally, the gray areas in the figures represent the $z$ range over which the lattice data are available.
• Figure 3: The figures display the speed of sound and specific heat, with red solid lines representing our HoloNet results and the error bars indicating the lattice data bazavov2014equation. As anticipated, these quantities exhibit excellent agreement once the model accurately satisfies the EoS.
• Figure 4: EoS comparison between neural network results and LQCD data bazavov2014equation. The error bars represent the LQCD data, while the solid lines represent the HoloNet model results. The blue curve corresponds to the trace anomaly $\epsilon-3p$, the yellow to the entropy density $s$, the green to the pressure $p$, and the red to the energy density $\epsilon$. The entropy density is used in training the HoloNet model. Since these are non-independent thermodynamic variables, the pressure, trace anomaly, and energy density generally agree with the lattice data automatically.
• Figure 5: Baryon number susceptibility comparison between HoloNet result and LQCD data borsanyi2012fluctuations. The black dots represent the LQCD results, with error bars omitted due to their negligible size. The red solid line represents the neural network model results. The optimized model exhibits a high level of quantitative agreement with the lattice results.