Deep Neural Networks: A Formulation Via Non-Archimedean Analysis

A new class of deep neural networks (DNNs) has been introduced, featuring multilayered tree-like architectures based on the ring of integers of non-Archimedean local fields. These DNNs are demonstrated to be robust universal approximators for real-valued functions on these rings, as well as for real-valued square-integrable functions on the unit interval. This development is significant for practitioners as it offers a novel mathematical framework that could enhance the robustness and versatility of neural network architectures in various applications.