# Machine Learning Holographic Mapping by Neural Network Renormalization Group

## TL;DR

This paper introduces a neural network-based method to automatically design holographic mappings for interacting field theories, enabling the extraction of emergent bulk geometries from boundary data.

## Contribution

It presents a universal neural network renormalization group approach to construct optimal holographic mappings for complex interacting theories.

## Key findings

- Successfully applied to 2D $\,\phi^4$ theory at criticality
- Emergent bulk geometry matches 3D hyperbolic space
- Demonstrates neural networks can optimize holographic dualities

## Abstract

The exact holographic mapping (EHM) provides an explicit duality map between a conformal field theory (CFT) configuration and a massive field propagating on an emergent classical geometry. However, designing the optimal holographic mapping is challenging. Here we introduce the neural network renormalization group as a universal approach to design generic EHM for interacting field theories. Given a field theory action, we train a flow-based hierarchical deep generative neural network to reproduce the boundary field ensemble from uncorrelated bulk field fluctuations. In this way, the neural network develops the optimal renormalization group transformations. Using the machine-designed EHM to map the CFT back to a bulk effective action, we determine the bulk geodesic distance from the residual mutual information. We apply this approach to the complex $\phi^4$ theory in two-dimensional Euclidian spacetime in its critical phase, and show that the emergent bulk geometry matches the three-dimensional hyperbolic geometry.