TITLE: Neural networks over gadgets other than $\mathbb{R}$
QUESTION [9 upvotes]: Recently, I learned that neural networks (NN) can be defined over fields other than $\mathbb{R}$: for example, Khrennikov and Tirozzi wrote a paper in 1999 (!) on $p$-adic neural networks, or neural networks over $p$-adic fields. It seems that there are some applications towards $p$-adic dynamical systems.
Are there similar lines of work, building NNs over gadgets other than $\mathbb{R}$? For example, some sort of arithmetic neural network?
In categorical terms, consider the category $\mathbf{CartSp}$ of cartesian spaces ($\mathbb{R}^n$); one can say that a neural network learns morphisms in this category. My question could be formulated as: over which categories could NNs be used to learn morphisms?

REPLY [7 votes]: Studies of neural networks that are more general than $\mathbb{R}^n\mapsto\mathbb{R}^k$ include

Deep Complex Networks (2017)
Deep Quaternion
Networks (2017)
P-Adic
Neural Networks (2004)
Deep
Learning on Lie Groups (2017)

A general overview is provided in Neural Networks on Groups (2019).
One general thing to keep in mind in this context is that the training of the network by steepest descent will rely on continuous differentiability, which not all implementations provide.