Holographic entanglement entropy, Wilson loops, and neural networks

TL;DR

This paper introduces a neural network framework to reconstruct bulk holographic geometries from boundary entanglement entropy and Wilson loop data, achieving high accuracy without relying on closed-form derivatives.

Contribution

It presents a novel semi-analytical and neural network-based method to invert holographic data, resolving degeneracies by combining entanglement entropy and Wilson loop measurements.

Findings

Achieves 1.7% accuracy in recovering the blackening factor in AdS-Schwarzschild backgrounds.

Demonstrates that entanglement entropy alone determines only the spatial metric.

Incorporates Wilson loop data to uniquely determine the timelike metric component.

Abstract

We apply artificial neural networks to the holographic inverse problem, reconstructing bulk geometry from boundary entanglement entropy by using the Ryu--Takayanagi area functional as a differentiable loss. Validated on the AdS-Schwarzschild background, this approach recovers the blackening factor to 1.7% accuracy. For finite-density backgrounds like the Gubser--Rocha model, we demonstrate that strip entanglement entropy determines only the spatial metric. We resolve this exact one-function degeneracy by incorporating holographic Wilson loop data, which couples to the timelike metric. We present a semi-analytical inversion combining Bilson's and Hashimoto's formulas, alongside a general three-network variational method minimizing the combined area and Nambu--Goto actions. The neural network achieves sub-0.2% accuracy for both metric functions without closed-form derivative relations, establishing a flexible framework for integrating multiple holographic observables.