将神经网络权重升级为张量，发现大宽限下出现费米子关联规律。
The neural networks with tensor weights and emergent fermionic Wick rules in the large-width limit
- 用张量权重替代标量权重，构建复杂值神经网络
- 大宽限下输出相关函数满足费米子的维克规则，可表为行列式
- 为量子场论类比神经网络提供费米子结构新路径
本文在神经网络量子场论（NN-QFT）框架下，研究具有张量权值的复值神经网络（CVNNs）。对于标准的标量权重CVNN，推导出生成泛函并识别出无限宽限下的精确高斯过程及其对应的有效量子态。当最后一层权重升级为克莱夫代数张量时，网络输出变为复矩阵值，大宽限下输出相关函数中诱导出类费米子符号结构。我们证明，在无限宽限极限下，具有相等数量$f^†$和$f$的关联函数遵循费米子维克规则，可表示为由标量欧几里得核$S(x,y)=\langle f^†(x)f(y)\rangle$构建的行列式。这在欧几里得关联函数和费曼规则层面实现了对NN-QFT的符号结构扩展，尽管尚未建立网络参数的微观格拉斯曼路径积分表示。分析表明，该框架已超越纯玻色子高斯场，为在神经架构中编码类费米子对称性提供了可能途径。
原文摘要 · Abstract (English)
In this paper, we study complex-valued neural network (CVNNs) with tensor-valued hidden-to-output weights within the framework of neural-network quantum field theory (NN-QFT). For standard CVNNs with scalar weights, we derive the generating functional and identify the exact Gaussian process that arises in the infinite-width limit, together with its associated effective quantum state. When the last-layer weights are promoted to Clifford-algebra-valued tensors, the network output becomes complex matrix-valued, and a fermion-like sign structure in the large-width correlation functions of the network output is induced. We show that, in the infinite-width limit, correlators with equal numbers of $f^†$ and $f$ obey fermionic Wick rules and can be written as determinants built from a scalar Euclidean kernel $S(x,y)=\langle f^†(x)f(y)\rangle$. This provides a sign-structured extension of NN-QFT at the level of Euclidean correlators and Feynman rules, even though a microscopic Grassmann path integral representation for the network parameters has not yet been constructed. Our analysis thus pushes the NN-QFT correspondence beyond purely bosonic Gaussian fields and suggests a possible route to encoding fermion-like symmetries in neural architectures for QFT correspondence.
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