Near-optimal deep neural network approximation for Korobov functions with respect to L^p and H¹ norms

This paper derives the optimal rate of approximation for Korobov functions with deep neural networks in the high dimensional hypercube with respect to L^p-norms and H¹-norm. Our approximation bounds are non-asymptotic in both the width and depth of the networks. The obtained approximation rates demonstrate a remarkable super-convergence feature, improving the existing convergence rates of neural networks that are continuous function approximators. Finally, using a VC-dimension argument, we show that the established rates are near-optimal.