Constructive Neural Network Field Theory: ϕ24 in Finite Volume
Abstract
Neural network field theory specifies a Euclidean field theory by a network architecture and a density on its parameters, so that correlation functions are integrals over those parameters. We introduce constructive neural network field theory and use it to realize finite-volume scalar ϕ42 as a limit of network measures. For each dyadic momentum shell of the free propagator, the field is given a separate set of neurons with Gaussian coefficients, uniform phases, and frequencies drawn from the shell's spectral density. Once the frequencies and phases are fixed, integrating out one shell's coefficients is a finite-dimensional integral against a Gaussian density reweighted by the Wick-ordered quartic interaction. For widths that grow fast enough with the shell index and almost every draw of the frequencies and phases, the reweighted density is uniformly log-concave for every shell above a threshold, at any coupling in two and three dimensions, and the fluctuations of those shell fields are held to the free-field scale. In two dimensions, an induction over scales shows that shell j contributes O(2−2jj3) to the log partition function, and the total-variation distance between marginals at successive cutoffs is of the same order. Because these contributions are summable, the ultraviolet cutoff can be removed for any suitable fixed sequence of widths. Almost surely, the ultraviolet and infinite-width limits commute, and in either order the joint laws of finitely many smeared fields converge to those of the standard finite-volume ϕ42 measure. The method of parametrizing a field via a network organized into momentum shells and integrating out one shell at a time is not specific to ϕ42, so we expect it to extend to other theories.
Related
- Neural Finite Volume and Mixture-of-Experts Physics-Informed Networks for the Isentropic Cargo–LeRoux Model: An Exact-Solution-Grounded Comparative Study
- Why Multi-Layer Message Passing Works: Completeness Theory for Graph Neural Network Interatomic Potentials
- Stable time-stepping via residual minimization: finite-element and neural-network approximations for transient parabolic problems
- Sparsely connected neural network representation of Lagrange finite element function
- What Neural Network Field Theory Can and Cannot Realise on a Computer
BibTeX
@misc{frank2026constructive,
title = {{Constructive Neural Network Field Theory: $ϕ_2^4$ in Finite Volume}},
author = {Samuel Frank},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.00453},
eprint = {2610.00453},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.00453},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Constructive Neural Network Field Theory: ϕ_2^4 in Finite Volume
AU - Samuel Frank
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.00453
UR - https://arxiv.org/abs/2610.00453
AB - Neural network field theory specifies a Euclidean field theory by a network architecture and a density on its parameters, so that correlation functions are integrals over those parameters. We introduce constructive neural network field theory and use it to realize finite-volume scalar ϕ^4_2 as a limit of network measures. For each dyadic momentum shell of the free propagator, the field is given a separate set of neurons with Gaussian coefficients, uniform phases, and frequencies drawn from the shell's spectral density. Once the frequencies and phases are fixed, integrating out one shell's coefficients is a finite-dimensional integral against a Gaussian density reweighted by the Wick-ordered quartic interaction. For widths that grow fast enough with the shell index and almost every draw of the frequencies and phases, the reweighted density is uniformly log-concave for every shell above a threshold, at any coupling in two and three dimensions, and the fluctuations of those shell fields are held to the free-field scale. In two dimensions, an induction over scales shows that shell j contributes O(2^−2jj^3) to the log partition function, and the total-variation distance between marginals at successive cutoffs is of the same order. Because these contributions are summable, the ultraviolet cutoff can be removed for any suitable fixed sequence of widths. Almost surely, the ultraviolet and infinite-width limits commute, and in either order the joint laws of finitely many smeared fields converge to those of the standard finite-volume ϕ^4_2 measure. The method of parametrizing a field via a network organized into momentum shells and integrating out one shell at a time is not specific to ϕ^4_2, so we expect it to extend to other theories.
ER -