ReLU神经网络对Korobov函数逼近误差可达近最优超收敛率。
Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions
- 基于稀疏网格有限元与位提取技术，构建高效逼近方法。
- 在L_p和W^1_p范数下，逼近误差分别达2m和2m-2阶。
- 揭示神经网络可有效突破维数灾难，适合高维函数逼近任务。
本文研究了ReLU神经网络对Korobov函数在$L_p$和$W^1_p$范数下的逼近误差。针对每方向具有$m$阶混合导数的函数，我们推导出网络宽度与深度下，$L_p$范数误差为$2m$阶、$W^1_p$范数误差为$2m-2$阶的近乎最优超逼近界。分析借助稀疏网格有限元与位提取技术，结果优于经典的一阶$L_ ty$与$H^1$误差界，表明神经网络表达能力基本不受维数灾难影响。
原文摘要 · Abstract (English)
This paper examines the $L_p$ and $W^1_p$ norm approximation errors of ReLU neural networks for Korobov functions. In terms of network width and depth, we derive nearly optimal super-approximation error bounds of order $2m$ in the $L_p$ norm and order $2m-2$ in the $W^1_p$ norm, for target functions with $L_p$ mixed derivative of order $m$ in each direction. The analysis leverages sparse grid finite elements and the bit extraction technique. Our results improve upon classical lowest order $L_\infty$ and $H^1$ norm error bounds and demonstrate that the expressivity of neural networks is largely unaffected by the curse of dimensionality.
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