Holographic flavour and neural networks

TL;DR

This work shows that physics‑informed neural networks can directly minimize the regularized Dirac–Born–Infeld action to obtain D7-brane embeddings in holography, bypassing the need to solve nonlinear Euler–Lagrange equations. The method reproduces known results for magnetic catalysis and meson melting, and extends to learning a one‑parameter family of embeddings and performing inverse problems to reconstruct bulk geometry from field theory data. By combining data-driven losses with physical losses, the approach yields embeddings, free-energy curves, and even derivatives with respect to parameters, offering a flexible tool for exploring holographic setups where solving the EOMs is challenging. While forward tasks may be slower than optimized ODE solvers, the framework excels at inverse problems and branch tracking, with potential extensions to more complex brane systems and metric reconstructions.

Abstract

In holography, flavour probe branes are used to introduce fundamental matter to the AdS/CFT correspondence. At a technical level, the probes are described by extremizing the DBI action and solving the Lagrange-Euler equations of motion. I report on applications of artificial neural networks that allow direct minimization of the regularized DBI action (interpreted as a free energy) without the need to derive and solve the equations of motion. I consider, as examples, magnetic catalysis of chiral symmetry breaking and the meson melting phase transition in the D3/D7 holographic set-up. Finally, I provide a framework which allows the simultaneous learning of the embeddings and the relevant aspects of the dual geometry based on field theory data.

Figures (13)

• Figure 1: Architecture of the ANN used to learn the profile $L(\rho)$. The parameter $\rho$ is in the interval $[0, \rho_{max}]$ and $L(\rho) = m + (\rho_{max} - \rho) g(\rho)$
• Figure 2: Plots of $L$ versus $\rho$ for various mass parameter $m$. One can see that the ANN produces smooth functions.
• Figure 3: Comparison of the free energy functional $I_{D7}$ (denoted by $F$ in the plot) between the numerical solution obtained by solving the equations of motion using shooting techniques and the free energy as given by the loss of the training process. One can observe an excellent agreement.
• Figure 4: Plots of $L$ versus $\rho$ for various mass parameter $m$. The curves inside the horizon are not physical and do not contribute to the gradient during training. The red rectangle represent the area of the first order phase transition pattern.
• Figure 5: Plot of the free energy functional (\ref{['ID7-hot']}) as a function of the bare mass parameter $m$. The agreement between the two approaches is excellent.