Error bounds for approximation with neural networks

In this paper we prove convergence rates for the problem of approximating functions f by neural networks and similar constructions. We show that the rates are the better the smoother the activation functions are, provided that f satisfies an integral representation. We give error bounds not only in Hilbert spaces but in general Sobolev spaces. Finally, we apply our results to a class of perceptrons and present a sufficient smoothness condition on f guaranteeing the integral representation.