  This paper establishes the nearly optimal rate of approximation for deep
neural networks (DNNs) when applied to Korobov functions, effectively
overcoming the curse of dimensionality. The approximation results presented in
this paper are measured with respect to $L_p$ norms and $H^1$ norms. Our
achieved approximation rate demonstrates a remarkable "super-convergence" rate,
outperforming traditional methods and any continuous function approximator.
These results are non-asymptotic, providing error bounds that consider both the
width and depth of the networks simultaneously.
