发表机构
Northeastern University; The NSF Institute for Artificial Intelligence and Fundamental Interactions (IAIFI)（东北大学; 美国国家科学基金会人工智能与基本相互作用研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出构造性神经网络场论，通过动量壳层网络参数化实现有限体积 $\phi^4_2$ 场论，证明紫外与无限宽极限可交换，并期望方法可推广至其他理论。
AI 中文摘要
神经网络场论通过网络架构及其参数上的密度来指定一个欧几里得场论，使得关联函数成为对这些参数的积分。我们引入构造性神经网络场论，并利用它实现有限体积标量 $\phi^4_2$ 作为网络测度的极限。对于自由传播子的每个二进动量壳层，场被赋予一组独立的神经元，这些神经元具有高斯系数、均匀相位，以及从该壳层谱中抽取的频率。一旦频率和相位固定，对某个壳层系数的积分就是关于高斯密度的有限维积分，该密度由 Wick 有序四次相互作用重新加权。在二维和三维中，对于任何耦合常数，阈值以上的每个壳层的重新加权密度都是一致对数凹的，并且这些壳层场的涨落被控制在自由场尺度上。在二维中，对于几乎每一次频率和相位的抽取，按尺度归纳表明，壳层 $j$ 对对数配分函数的贡献为 $O(2^{-2j}j^3)$，并且相继截断处边缘分布之间的全变差距离具有相同的阶数。由于这些贡献是可求和的，对于任何随壳层指标增长足够快的固定宽度序列，紫外截断都可以被移除。几乎必然地，紫外极限和无限宽极限可交换，并且无论以何种顺序，有限多个涂抹场的联合分布都收敛到标准有限体积 $\phi^4_2$ 测度的联合分布。这种通过按动量壳层组织的网络来参数化场并一次积分一个壳层的方法并非 $\phi^4_2$ 所特有，因此我们预期它可以推广到其他理论。
英文摘要
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 $O(2^{-2j}j^3)$ 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.
Comments44 pages. v2: structure and style revised, results unchanged
