# Deep Neural Networks: A Formulation Via Non-Archimedean Analysis

arXiv: https://arxiv.org/abs/2402.00094 · HTML: https://mlsanity.com/paper/2402.00094/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 42 references that are in the map.
- First posted on arXiv 31 Jan 2024.

## Builds on (most cited first, 8 of 8)

- [Universality of Deep Convolutional Neural Networks](https://mlsanity.com/paper/1805.10769/) (2018)
- [The Principles of Deep Learning Theory](https://mlsanity.com/paper/2106.10165/) (2021)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [The staircase property: How hierarchical structure can guide deep learning](https://mlsanity.com/paper/2108.10573/) (2021)
- [Approximation spaces of deep neural networks](https://mlsanity.com/paper/1905.01208/) (2019)
- [Unified field theoretical approach to deep and recurrent neuronal networks](https://mlsanity.com/paper/2112.05589/) (2021)
- [p-Adic Statistical Field Theory and Deep Belief Networks](https://mlsanity.com/paper/2207.13877/) (2022)
- [Hierarchical Wilson-Cowan Models and Connection Matrices](https://mlsanity.com/paper/2305.00120/) (2023)

## Built on by (most cited first, 1 of 1)

- [Non-Archimedean Polydisc Spaces and Applications to Optimisation](https://mlsanity.com/paper/2606.07782/) (2026)

## Abstract (arXiv)

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified using numbers from the ring of integers of non-Archimdean local fields. These rings have a natural hierarchical organization as infinite rooted trees. Natural morphisms on these rings allow us to construct finite multilayered architectures. The new DNNs are robust universal approximators of real-valued functions defined on the mentioned rings. We also show that the DNNs are robust universal approximators of real-valued square-integrable functions defined in the unit interval.