Constructive Neural Network Field Theory: $φ_2^4$ 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 $φ<sup>4_2$ 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 spectrum. 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. At any coupling in two and three dimensions, the reweighted density is uniformly log-concave for every shell above a threshold, and the fluctuations of those shell fields are held to the free-field scale. In two dimensions, for almost every draw of the frequencies and phases, an induction over scales shows that shell $j$ contributes $O(2<sup>{-2j}j<sup>3)$ 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 fixed sequence of widths that grows fast enough with the shell index. 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 $φ<sup>4_2$ 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 $φ<sup>4_2$, so we expect it to extend to other theories.