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 \((-1,1)\) . We assume that the given source term and reaction coefficient are analytic in \([-1,1]\) . 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 \(\varepsilon \in (0,1]\) 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.
