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<p>We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified by using numbers in the ring of integers of non-Archimedean 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.</p>

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Deep Neural Networks: A Formulation via Non-Archimedean Analysis

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  • 出版模式：

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  • 发表日期：

    2026年7月27日

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  • W. A. Zúñiga-Galindo

摘要

We introduce a new class of deep neural networks (DNNs) with multilayered tree-like architectures. The architectures are codified by using numbers in the ring of integers of non-Archimedean 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.