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.

翻译：本文提出了一类具有多层树状架构的新型深度神经网络（DNNs）。此类架构通过非阿基米德局部域的整数环中的数字进行编码。这些整数环具有无限根树的自然分层组织结构。基于环上的自然同态映射，我们得以构建有限层的多层架构。新提出的DNNs能够稳健地泛逼近定义在上述整数环上的实值函数。我们进一步证明，此类DNNs同样能够稳健地泛逼近定义在单位区间上的实值平方可积函数。