Some super-approximation rates of ReLU neural networks for Korobov functions
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.
Fields of Study & Subjects
Neural Networks and Applications
Image and Signal Denoising Methods
Model Reduction and Neural Networks
Publication details
- Journal / source
- Communications in Mathematical Sciences
- Volume
- 24
- Issue
- 8
- Pages
- 2159–2186
- Publication date
- 1 January 2026
- Language
- EN