Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms
Functional Analysis
2019-02-22 v1 Machine Learning
Abstract
We analyze approximation rates of deep ReLU neural networks for Sobolev-regular functions with respect to weaker Sobolev norms. First, we construct, based on a calculus of ReLU networks, artificial neural networks with ReLU activation functions that achieve certain approximation rates. Second, we establish lower bounds for the approximation by ReLU neural networks for classes of Sobolev-regular functions. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.
Keywords
Cite
@article{arxiv.1902.07896,
title = {Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms},
author = {Ingo Gühring and Gitta Kutyniok and Philipp Petersen},
journal= {arXiv preprint arXiv:1902.07896},
year = {2019}
}