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There is probably a ton of isomorphism between different models. It may come down to what is easiest to understand and fastest to implement in code.

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See also:

A visual proof that neural networks can approximate any function

https://news.ycombinator.com/item?id=19708620

So a "neural network" is actually a type of parameterized mathematical function that can be fit to any curve including higher dimensional surfaces, etc.?

Yes.

That's correct. See https://en.wikipedia.org/wiki/Universal_approximation_theore... for more details

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