FYI, there are actually many algorithms going back longer than the neural network algorithm that have been proven to be a universal function approximator. Neural networks are certainly not the only and not the first to do so. There are quite a few that are actually much more appropriate for many cases than a neural network.
For your interest, Taylor Series are not universal function approximators - the Taylor Series around 0 for
f(x) = e^(-1/x^2) if x != 0 else 0
is identically zero (all partial derivatives are 0 at 0) but the function is clearly not identically zero. So the radius of convergence for this Taylor series is infinite but it only equals the approximated function at one point.
I'm sure there are some conditions you can put on f to make the Taylor Series a UFA but it's been quite a while since I did any real analysis so I have forgotten!
Doesn't detract from the overall point though that there are UFAs that are not neural nets. I should say that I don't know what the precise definition of a UFA really is, but I assume you have to have more than equality at one point.
Taylor series work on differentiable intervals. You specifically chose a function and interval where this is not true. Of course it will not be a good approximation.
This area is covered by non-parametric statistics more generally. There are many other methods to non-parametrically estimate functions (that satisfy some regularity conditions). Tree-based methods are one family of such methods, and the consensus still seems to be that they perform better than neural networks on tabular data. For example:
Newtons Method approximates square roots. Its useful if you want to approximate something like that without pulling in the computational power required of NN.
I think the problem to solve is more like : given a set of inputs and outputs, find a function that gives the expected output for each input [1]. This is like Newton's method on a higher order ;-). One can find such a tool in Squeak or Pharo Smalltalk, IIRC.
