# Intuition behind universal approximation theorem

I can imagine a lot of math's theorems and laws, but I can not imagine the "process" universal approximation theorem is talking about.

1) Can you explain in simple words the proof of universal approximation theorem?

2) Why neural networks with 1+ hidden layers are working?

3) Are neural networks analog of math's series?

• In Wikipedia's terminology, you can approximately expand a continuous real-valued function on the unit hypercube into a finite linear combination of functions of the form $\varphi \circ a_i$ where $a_i$ are affine functions and $\varphi$ can be taken to be a single, rather "nice", function. If my limited understanding is correct, these $a_i$ form the "hidden layer" of the network. And yes, this expansion of a "generic" function into a sum of "nice" functions is similar to expansion into power series, Fourier series, or wavelet series. – Ian Mar 20 '17 at 12:31
• @Ian I need the proof – Dmitry Nalyvaiko Mar 20 '17 at 12:35
• Surely it can be looked up? Indeed the Wikipedia article gives 5 references... – Ian Mar 20 '17 at 12:36
• @Ian my brain very poorly delves into the scientific terminology, that's why I ask here in the hope of getting an understandable explanation – Dmitry Nalyvaiko Mar 20 '17 at 12:39