Near-optimal deep neural network approximation for Korobov functions with respect to Lp 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 $H^1$-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.