$p$-Adic Polynomial Regression as Alternative to Neural Network for Approximating $p$-Adic Functions of Many Variables

A method for approximating continuous functions ${\mathbb{Z}}_p^n\to {\mathbb{Z}}_p$ by a linear superposition of continuous functions ${\mathbb{Z}}_p\to {\mathbb{Z}}_p$ is presented and a polynomial regression model is constructed that allows approximating such functions with any degree of accuracy. A physical interpretation of such a model is given and possible methods for its training are discussed. The proposed model can be considered as a simple alternative to possible $p$-adic models based on neural network architecture.