We propose an efficient, deterministic algorithm for constructing exponentially convergent deep neural network (DNN) approximations of multivariate, analytic maps f:[−1,1]K→R. We address in particular networks with the rectified linear unit (ReLU) activation function. Similar results and proofs apply for many other popular activation functions. The algorithm is based on collocating f in deterministic families of grid points with small ...

We prove exponential expressivity with stable ReLU Neural Networks (ReLU NNs) in H1(Ω) for weighted analytic function classes in certain polytopal domains Ω, in space dimension d=2,3. Functions in these classes are locally analytic on open subdomains D⊂Ω, but may exhibit isolated point singularities in the interior of Ω or corner and edge singularities at the boundary ∂Ω. The exponential expression rate bounds proved here imply uniform ...

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]. We establish expression rate bounds in Sobolev norms in terms of the NN size which are uniform with respect to the singular perturbation ...