AI 中文总结
本文提出神经网络函数空间临界性的威尔逊诠释，将有限宽度修正视为非高斯相互作用微扰，揭示有限神经网络临界结构源于有限宽度效应，为研究神经网络可训练性等提供理论基础。
AI 中文摘要
神经网络不仅可作为参数化计算模型，也可作为函数上的概率分布进行研究。本文在神经网络-量子场论对应关系中提出临界性的威尔逊诠释，将无限宽度高斯过程极限视为自由场定点，有限宽度修正视为产生非高斯相互作用的微扰。由此，我们不再将偏离无限宽度视为小近似误差，而是将其视为相互作用、复杂性、表达能力及类相行为进入神经网络函数空间的过程。宽度、深度、激活非线性、初始化方差及训练动态被视为改变网络系综有效作用量的控制参数。临界状态下，高阶连通关联函数变得不可忽略，有限宽度算子成为相关或边际标度因子，函数分布对依赖标度的结构敏感。我们可将过参数化视为抑制相互作用项的关系，发现有限神经网络的临界结构由有限宽度效应决定。该框架为通过威尔逊定点、微扰及临界曲面研究神经网络函数的可训练性、泛化性及架构普适性提供了理论基础。
英文摘要
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.
Comments30 pages, 16 figures, presented at the 3rd International Conference Mathematical Analysis and Applications in Science and Engineering (ICMASC) 2026 Conference, Portugal
