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

This paper examines the Lₚ and W¹ₚ 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 Lₚ norm and order $2m-2$ in the W¹ₚ norm, for target functions with Lₚ 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_∞ and H¹ norm error bounds and demonstrate that the expressivity of neural networks is largely unaffected by the curse of dimensionality.