# On the approximation of rough functions with deep neural networks

arXiv: https://arxiv.org/abs/1912.06732 · HTML: https://mlsanity.com/paper/1912.06732/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Dec 2019.

## Built on by (most cited first, 3 of 3)

- [GPT-PINN: Generative Pre-Trained Physics-Informed Neural Networks toward non-intrusive Meta-learning of parametric PDEs](https://mlsanity.com/paper/2303.14878/) (2023)
- [Error analysis for deep neural network approximations of parametric hyperbolic conservation laws](https://mlsanity.com/paper/2207.07362/) (2022)
- [Learning WENO for entropy stable schemes to solve conservation laws](https://mlsanity.com/paper/2403.14848/) (2024)

## Abstract (arXiv)

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.