Hello.

Let's say I have a set of input vectors $I = \{\mathbf{x_1}, \dots, \mathbf{x_k}\} \subset \mathcal{R}^m$ and a set of output vectors $O = \{\mathbf{y_1}, \dots, \mathbf{y_k}\} \subset \mathcal{R}^n$, and I want to obtain a mapping $f : \mathcal{R}^m \to \mathcal{R}^n$ such that

$$ f(\mathbf{x_i}) = \mathbf{y_i} + \epsilon_i, \forall i \in \{1, \dots, k\}$$

where $\epsilon_i$ is small, and this mapping should be continuous at least around the input/output pairs.

There are many ways of doing so.

If we suppose that the input/output pairs won't change, what are the advantages of using an Artificial Neural Network over other methods to approximate functions?

Thanks.