Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions

• URL: http://arxiv.org/abs/2507.10345v1
• Date: Mon, 14 Jul 2025 14:48:47 GMT
• Title: Some Super-approximation Rates of ReLU Neural Networks for Korobov Functions
• Authors: Yuwen Li, Guozhi Zhang,
• Abstract summary: We derive nearly optimal super-approximation error bounds of order $2m$ in the $L_p$ norm and order $2m-2$ in the $WLUp$ norm for target functions with $L_p$ mixed of order $m$ in each direction.<n>Our results show that the expressivity of neural networks is largely unaffected by the curse of dimensionality.
• Score: 3.2228025627337864
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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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