用单个p进制特征函数构建神经网络，实现p进制通用逼近。
$p$-adic Character Neural Network
- 以p进制整数群上的单个内射特征函数作为激活函数
- 证明了该网络可逼近任意连续函数，等价于多项式方程在模p^k下的可解性
- 为基于代数结构的神经网络设计提供新思路，适合数学与理论机器学习研究者
我们提出一种新的p进制神经网络框架。不同于Albeverio、Khrennikov和Tirrozi使用一组由精度超参数索引的特征函数作为激活函数的原始方法，我们采用拓扑阿贝尔群ℤₚ（p进制整数）上的单个内射p进制特征函数作为激活函数。我们证明了该形式的p进制神经网络具有p进制通用逼近性质，并将其简化为在模p的幂次有限环上多项式方程的可解性问题。
原文摘要 · Abstract (English)
We propose a new frame work of $p$-adic neural network. Unlike the original $p$-adic neural network by S.\ Albeverio, A.\ Khrennikov, and B.\ Tirrozi using a family of characteristic functions indexed by hyperparameters of precision as activation functions, we use a single injective $p$-adic character on the topological Abelian group $\mathbb{Z}_p$ of $p$-adic integers as an activation function. We prove the $p$-adic universal approximation theorem for this formulation of $p$-adic neural network, and reduce it to the feasibility problem of polynomial equations over the finite ring of integers modulo a power of $p$.
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