On the approximation of rough functions with deep neural networks

TL;DR

This paper demonstrates that the ENO interpolation method can be represented as a deep ReLU neural network, enabling high-order approximation of rough functions with applications in nonlinear conservation laws and data compression.

Contribution

It proves that ENO interpolation can be implemented as a deep neural network, bridging classical approximation techniques with modern deep learning frameworks.

Findings

Neural networks can replicate high-order ENO interpolation.

The approach achieves excellent approximation of solutions to nonlinear conservation laws.

Numerical tests confirm the effectiveness in data compression.

Abstract

Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural networks, including its high-order accuracy at approximating Lipschitz functions. Numerical tests for the resulting neural networks show excellent performance for approximating solutions of nonlinear conservation laws and at data compression.