  Rational and neural network based approximations are efficient tools in
modern approximation. These approaches are able to produce accurate
approximations to nonsmooth and non-Lipschitz functions, including multivariate
domain functions. In this paper we compare the efficiency of function
approximation using rational approximation, neural network and their
combinations. It was found that rational approximation is superior to neural
network based approaches with the same number of decision variables. Our
numerical experiments demonstrate the efficiency of rational approximation,
even when the number of approximation parameters (that is, the dimension of the
corresponding optimisation problems) is small. Another important contribution
of this paper lies in the improvement of rational approximation algorithms.
Namely, the optimisation based algorithms for rational approximation can be
adjusted to in such a way that the conditioning number of the constraint
matrices are controlled. This simple adjustment enables us to work with high
dimension optimisation problems and improve the design of the neural network.
The main strength of neural networks is in their ability to handle models with
a large number of variables: complex models are decomposed in several simple
optimisation problems. Therefore the the large number of decision variables is
in the nature of neural networks.
