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：我们提出了一类具有多层树状架构的新型深度神经网络。这些架构通过非阿基米德局部域整数环中的数字进行编码。该整数环具有无限有根树的天然层级化组织结构，其上的自然态射使我们能够构建有限多层架构。此类新型深度神经网络是定义在上述环上的实值函数的鲁棒通用逼近器。我们还证明了，这类深度神经网络亦是定义在单位区间上的实值平方可积函数的鲁棒通用逼近器。