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.
[1] https://stackoverflow.com/questions/1539286/create-a-functio...