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.

翻译：本文建立了深度神经网络应用于Korobov函数时的近最优逼近速率，有效克服了维度灾难。本文给出的逼近结果以$L_p$范数和$H^1$范数为度量。所取得的逼近速率展现出显著的"超收敛"特性，优于传统方法和任何连续函数逼近器。这些结果是非渐近的，所给出的误差界同时考虑了网络的宽度和深度。