Constructive approximate interpolation by neural networks in the metric space

In this paper, we construct two types of feed-forward neural networks (FNNs) which can approximately interpolate, with arbitrary precision, any set of distinct data in the metric space. Firstly, for analytic activation function, an approximate interpolation FNN is constructed in the metric space, and the approximate error for this network is deduced by using Taylor formula. Secondly, for a bounded sigmoidal activation function, exact interpolation and approximate interpolation FNNs are constructed in the metric space. Also the error between the exact and approximate interpolation FNNs is given.