关于Sobolev范数下Korobov函数的最优深度神经网络逼近
Optimal Deep Neural Network Approximation for Korobov Functions with respect to Sobolev Norms
AI总结：
该研究确立了深度神经网络对Korobov函数在$L_p$与$H^1$范数下的近最优非渐近逼近速率，实现超收敛并克服维数灾难，误差界同时考虑网络宽度与深度。
摘要：
本文确立了深度神经网络（DNNs）应用于Korobov函数时的近最优逼近速率，有效克服了维数灾难。本文给出的逼近结果是基于$L_p$范数和$H^1$范数衡量的。我们得到的逼近速率展现出显著的“超收敛”速率，优于传统方法及任意连续函数逼近器。这些结果是非渐近的，给出了同时考虑网络宽度和深度的误差界。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
