Learning the Inverse Ryu--Takayanagi Formula with Transformers

TL;DR

This work tackles the inverse problem of holographic entanglement entropy in AdS$_3$ by training a Transformer to reconstruct the bulk blackening function $f(z)$ from boundary entanglement data $S(\ell)$. By generating a large, noise-augmented dataset of BTZ-like geometries and using the Hamilton–Jacobi relation to convert $S(\ell)$ into $\ell(z_t)$, the model learns the inverse RT map in a data-driven, physics-agnostic manner. The results show accurate recovery of smooth and horizonless geometries, as well as robust predictions under a variety of entanglement-entropy perturbations (exponential, hyperbolic-tangent, power interpolation, and periodic boundary cases). This demonstrates the feasibility of data-driven holographic inversion in AdS$_3$ and highlights the potential for extending the approach to higher dimensions and more complex entanglement structures, enabling geometry inference from entanglement data in a scalable way.

Abstract

We study the inverse problem of holographic entanglement entropy in AdS$_3$ using a data-driven generative model. Training data consist of randomly generated geometries and their holographic entanglement entropies using the Ryu--Takayanagi formula. After training, the Transformer reconstructs the blackening function within our metric ansatz from previously unseen inputs. The Transformer achieves accurate reconstructions on smooth black hole geometries and extrapolates to horizonless backgrounds. We describe the architecture and data generation process, and we quantify accuracy on both $f(z)$ and the reconstructed $S(\ell)$. Code and evaluation scripts are available at the provided repository.

Figures (6)

• Figure 1: Examples of inputs and targets used in training. Left: blackening functions $f(z)$ with additive white noise $\eta$. Right: the corresponding $\ell(z_t)$ evaluated via Eq. \ref{['eq:ellzt']}. Here $z_h=1$, $z\in[0,1]$ with $\Delta z=0.01$, $\mu=0$, and $\sigma=0.5$. Each figure shows four randomly drawn samples (blue, red, yellow, green), and colors correspond across figures.
• Figure 2: The four plots show outputs from the trained Transformer. The blue curves are the prediction of the AI model, and the orange curves are the true target data.
• Figure 3: Transformer predictions for the exponential perturbation form of the entanglement entropy. Left: predicted blackening functions $f(z)$ for $s\in\{0,0.24651,0.5,1,1.5,2\}$. Right: solid curves denote the input source $S'(\ell)$, while dotted curves show $S'(\ell)$ recomputed from the predicted $f(z)$.
• Figure 4: Left: predicted blackening functions $f(z)$ for $s\in\{0.5,1,1.5,2,2.5,3\}$. Right: solid curves denote the input source $S'(\ell)$, while dotted curves show $S'(\ell)$ recomputed from the predicted $f(z)$.
• Figure 5: Left: predicted blackening functions $f(z)$ for $s\in\{1.8, 2.3, 2.8, 3.3, 3.8, 4.3\}$. Right: solid curves denote the input source $S'(\ell)$, while dotted curves show $S'(\ell)$ recomputed from the predicted $f(z)$.