Universality of Neural Network Field Theory

TL;DR

The paper proves that any Euclidean quantum field theory valued in the space of tempered distributions $\mathcal{S}'(\mathbb{R}^d)$ admits a neural-network description with countably many parameters, formalized through the generalized quantum system (GQS) framework and a Borel-isomorphism to $\mathbb{R}^{\mathbb{N}}$. This universal NN-FT perspective is illustrated by a concrete realization of 2D Liouville theory on $S^2$, where the field is split into a zero mode and a Gaussian-free non-zero mode, and the $3$-point function is computed via Monte Carlo sampling of a deformed parameter density incorporating Gaussian multiplicative chaos, yielding agreement with the DOZZ formula to within a few percent. The approach offers a principled path to encode QFT observables as NN expectations and suggests practical strategies for constructing and training NN-FT representations. It also opens avenues for exploring other theories numerically within this unified framework, while posing questions about enforcing OS-like positivity and the physical interpretability of trained NN-FT architectures.

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.

Figures (3)

• Figure 1: The three-point function (\ref{['three_point_vertex']}) is computed for various values of $b$, with $\alpha_1 = \alpha_2$ fixed, and as $\alpha_3$ is varied. The NN-FT simulation is performed with $L = 30$ spherical harmonics and $10$ experiments with $50,000$ runs per experiment; error bars are shown measuring the variance across experiments but are too small to be seen. A UV regulator is introduced by pixelating the sphere at $100$ points of latitude and $200$ points of longitude. Exact results from the DOZZ formula are shown in solid curves and lie within all error bars.
• Figure 2: Distributional Nature of Large Width Liouville Network. The values of the truncated sum $\sum_{\ell=1}^{L} \sum_{m = - \ell}^{\ell} a_{\ell, m} Y_{\ell, m}$ are shown for increasing values of $L$. The magnitude of the sum is visualized both by the distance from the origin to a point on the surface, and by the color, with yellower tones corresponding to larger distances. As $L$ is increased, more "delta-function-like" spikes are seen to develop. This is because the coefficients $a_{\ell, m}$ do not decay sufficiently rapidly for the sum to converge to a smooth function, but the sum does converge almost surely to a distribution on the sphere.
• Figure 3: We compare the results of our NN-FT simulation to the DOZZ formula for values of the three-point function which are smaller than those shown in Figure \ref{['fig:dozz_prl_plot']}. Again, in each experiment, two values of the $\alpha_i$ are held fixed while the third is varied. Some of these data points are close to the Seiberg bound (\ref{['seiberg_bound']}), where the DOZZ formula generically has a zero and the techniques used to define Liouville correlators become more subtle. We draw $N = 75,000$ sample field configurations, pixelate the sphere at $128$ points of longitude and $64$ points of latitude, and perform $5$ iterations of importance sampling. For these values of parameters, the errors are larger and several of the DOZZ formula predictions do not lie within the error bars representing the standard deviation across experiments.