p-adic Character Neural Network

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 mathbbZp 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.