Universality of Neural Network Field Theory

TL;DR

This paper demonstrates that neural networks can universally represent any quantum field theory or probability distribution over distributions, and provides a concrete example with the Liouville theory, including numerical validation.

Contribution

It proves the universality of neural network representations for quantum field theories and offers a numerical example with Liouville theory matching theoretical predictions.

Findings

Neural networks can encode any quantum field theory.

Successfully realized 2D Liouville theory as a neural network.

Numerical three-point function matches the DOZZ formula.

Abstract

We prove that any quantum field theory, or more generally any probability distribution over tempered distributions in $\mathbb{R}^d$, admits a neural network description with a countable infinity of parameters. As an example, we realize the $2d$ Liouville theory as a neural network and numerically compute the three-point function of vertex operators, finding agreement with the DOZZ formula.