Neural networks for singular perturbations

Abstract We prove deep neural network (DNN for short) expressivity rate bounds for solution sets of a model class of singularly perturbed, elliptic two-point boundary value problems, in Sobolev norms, on the bounded interval (-1, 1) (- 1, 1). We assume that the given source term and reaction coefficient are analytic in -1, 1 - 1, 1. The expression rate bounds in Sobolev norms in terms of the NN size are robust, i. e. uniform with respect to the singular perturbation parameter (0, 1] ε ∈ (0, 1 ] for several classes of DNN architectures. In particular, ReLU NNs, spiking NNs, and tanh - and sigmoid-activated NNs. The latter activations can represent “exponential boundary layer solution features” explicitly, in the last hidden layer of the DNN, i. e. in a shallow subnetwork, and afford improved robust expression rate bounds in terms of the NN size. All DNN architectures allow robust exponential solution expression in so-called ‘energy’ as well as in ‘balanced’ Sobolev norms, for analytic input data.