# Error bounds for approximations with deep ReLU neural networks in $W^{s,p}$ norms

## TL;DR

This paper investigates how well deep ReLU neural networks can approximate Sobolev-regular functions in weaker Sobolev norms, providing both upper and lower bounds that are relevant for PDE numerical analysis.

## Contribution

It introduces new approximation rates for ReLU networks in Sobolev spaces and establishes fundamental lower bounds, extending approximation theory in this context.

## Key findings

- Constructed neural networks achieving specific approximation rates
- Established lower bounds for ReLU network approximation in Sobolev norms
- Extended approximation theory to PDE-relevant Sobolev regimes

## 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.