Source: Michael Nielsen blog, Jan 2016

universality tells us that neural networks can compute any function; and empirical evidence suggests that deep networks are the networks best adapted to learn the functions useful in solving many real-world problems.

One of the most striking facts about neural networks is that they can compute any function at all. That is, suppose someone hands you some complicated, wiggly function, f(x)f(x):

No matter what the function, there is guaranteed to be a neural network so that for every possible input, xx, the value f(x)f(x) (or some close approximation) is output from the network, e.g.:

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