# p-adic Character Neural Network

arXiv: https://arxiv.org/abs/2603.29905 · HTML: https://mlsanity.com/paper/2603.29905/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 20 references that are in the map.
- First posted on arXiv 31 Mar 2026.

## Builds on (most cited first, 3 of 3)

- [v-PuNNs: van der Put Neural Networks for Transparent Ultrametric Representation Learning](https://mlsanity.com/paper/2508.01010/) (2025)
- [Hierarchical Neural Networks, p-Adic PDEs, and Applications to Image Processing](https://mlsanity.com/paper/2406.07790/) (2024)
- [On the Local Ultrametricity of Finite Metric Data](https://mlsanity.com/paper/2408.07174/) (2024)

## Built on by (most cited first, 2 of 2)

- [p-adic Manifold Learning and Benchmark Tasks from Impartial Games](https://mlsanity.com/paper/2605.04374/) (2026)
- [Continuous Optimization for p-adic Models](https://mlsanity.com/paper/2609.25501/) (2026)

## Abstract (arXiv)

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.