Assuming that the analytical function is non-linear such as sin(x), I am interested to know if the mathematical evaluation of the function such as sin(x) by the expansion of Taylor series faster or slower then the evaluation of corresponding approximation of sin(x) via neural nets, with same inputs?
The complexity of computation is lower in neural nets due to its parallel nature so can the function be evaluated faster by neural nets? So to generalize it, is it always good to have expensive functions converted to their corresponding neural net counterparts so they can be evaluated faster for any range of inputs?
The accuracy of function approximation that you can obtain with neural nets is very poor (a neural net performs a clumsy interpolation). In addition, the evaluation of a single sigmoid function might be as costly as the evaluation of a sine function.
Never do that.
With a math coprocessor, even a simple polynomial approximation would be beaten by the built-in function.