# Towards a regularity theory for ReLU networks -- chain rule and global error estimates

## TL;DR

This paper develops a rigorous derivative concept for ReLU neural networks that satisfies the chain rule and provides a method to extend local approximation results to global estimates, enhancing understanding of neural network regularity.

## Contribution

It introduces a derivative framework compatible with the chain rule for ReLU networks and offers a technique to convert local approximation results into global estimates.

## Key findings

- A new derivative concept satisfying the chain rule for ReLU networks
- Method to extend local approximation results to global estimates
- Application to high-dimensional PDEs in deep learning

## Abstract

Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not applicable. We will consider a way of introducing a derivative for neural networks that admits a chain rule, which is both rigorous and easy to work with. In addition we will present a method of converting approximation results on bounded domains to global (pointwise) estimates. This can be used to extend known neural network approximation theory to include the study of regularity properties. Of particular interest is the application to neural networks with ReLU activation function, where it contributes to the understanding of the success of deep learning methods for high-dimensional partial differential equations.