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arXiv:2601.14453

Universality of Neural Network Field Theory

Christian Ferko, James Halverson, Aaron Mutchler

Published Jan 20, 2026• 0 citations
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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.

DOI
10.48550/arxiv.2601.14453

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Under mild assumptions, any quantum mechanical (QM) model or quantum field theory (QFT) admits a representation in terms of an ensemble of neural networks with countably many random parameters. We investigate the features of NN-QM and NN-FT models with finitely many parameters, such as a feedforward network of width N<∞N < \inftyN<∞. We find that, generically, such models must violate one of the properties of conventional Euclidean QFTs, such as reflection positivity or cluster decomposition. We present several complementary ways of understanding which features can and cannot be preserved at finite NNN, both in QM and in QFT.
A statistical ensemble of neural networks can be described in terms of a quantum field theory (NN-QFT correspondence). The infinite-width limit is mapped to a free field theory, while finite N corrections are mapped to interactions. After reviewing the correspondence, we will describe how to implement renormalization in this context and discuss preliminary numerical results for translation-invariant kernels. A major outcome is that changing the standard deviation of the neural network weight distribution corresponds to a renormalization flow in the space of networks.
Neural networks can be studied not only as parameterized computational models but also as probability distributions over functions. In this paper we develop a Wilsonian interpretation of criticality in the neural network-quantum field theory correspondence, treating the infinite width Gaussian-process limit as a free field fixed point and finite width corrections as perturbations that create non-Gaussian interactions. In this way, we do not see the departure from infinite width as a small approximation error but as the process by which interaction, complexity, expressivity and phase-like behaviour enter neural network function space. Width, depth, activation nonlinearity, initialization variance, and training dynamics are considered as control parameters that change the effective action of the network ensemble. The critical regime is when the higher-order connected correlation functions become non-negligible, and when the finite width operators become relevant or marginal scaling factors, and when the function distribution becomes sensitive to scale-dependent structure. There we can view overparameterization as suppressing a relationship of interacting terms and we find that the critical structure of finite neural networks is given by finite-width effects. This framework is a theoretical basis for studying trainability, generalization, and architectural universality of neural network functions through Wilsonian fixed points, perturbations, and critical surfaces.
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A bstract We use the embedding formalism to construct conformal fields in D dimensions, by restricting Lorentz-invariant ensembles of homogeneous neural networks in ( D + 2) dimensions to the projective null cone. Conformal correlators may be computed using the parameter space description of the neural network. Exact four-point correlators are computed in a number of examples, and we perform a 4D conformal block decomposition that elucidates the spectrum. In a non-unitary example the decomposition precisely matches OPE coefficients for the self-correlator, but not for the mixed correlator. In others, the analysis is facilitated by recent approaches to Feynman integrals. Generalized free CFTs are constructed using the infinite-width Gaussian process limit of the neural network, enabling a realization of the free boson. The extension to deep networks constructs conformal fields at each subsequent layer, with recursion relations relating their conformal dimensions and four-point functions. Numerical approaches are discussed.
Neural network conformal theories were first introduced in \cite{Halverson:2024axc} and exact two-, three- and four-point correlators were computed from an ensemble of infinite-width neural networks. In particular, the four-point correlator of any scalar operator satisfied crossing-symmetry and conformal block decomposition across different channels were in agreement without any constraint on moments. An infinite tower of degenerate operators with the same scaling dimension were found. In this note, we take a closer look at the structure of correlators and relation between operators. We also layout a recipe to construct the algebra of the theory.