On the approximation of functions by tanh neural networks

Abstract: We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function. These bounds provide explicit estimates on the approximation error with respect to the size of the neural networks. We show that tanh neural networks with only two hidden layers suffice to approximate functions at comparable or better rates than much deeper ReLU neural networks.