Error bounds in Sobolev norms for approximations with norm constrained ReLU neural networks
Recent studies have shown that smooth functions can be well approximated by ReLU neural networks with path norm constraint on the weights. We extend these results from uniform approximation to approximation in Sobolev norm. Specifically, we analyze how well Sobolev functions in $W^{n,p}$ can be approximated by neural networks with width $W$, depth $L$ and path norm bounded by $K$, when the approximation error is measured in the $W^{1,p}$-norm. For shallow networks with depth $L=1$, we derive the approximation error bound $\mathcal{O}(\max\{W^{-(n-1)/d}, K^{-(n-1)/(s-n)}\})$, when the smoothness index satisfies $n<s=(d+3)/2$ and the input is $d$-dimensional. For deep networks, we remove th...
| Authors | Xianjun Li, Yunfei Yang |
|---|---|
| Primary category | stat.ML |
| Published | 17 September 2026 |
| Last updated at source | 17 September 2026 |
Silytics shows public metadata from arXiv and links to the original. Information may be incomplete or out of date. Last refreshed 17 October 2026. Licenses and terms of the original apply to the work itself. Not financial, legal or professional advice.