# Depth separation for reduced deep networks in nonlinear model reduction: Distilling shock waves in nonlinear hyperbolic problems

arXiv: https://arxiv.org/abs/2007.13977 · HTML: https://mlsanity.com/paper/2007.13977/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jul 2020.

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

- [Depth separation beyond radial functions](https://mlsanity.com/paper/2102.01621/) (2021)
- [Machine learning moment closure models for the radiative transfer equation I: directly learning a gradient based closure](https://mlsanity.com/paper/2105.05690/) (2021)
- [A Low Rank Neural Representation of Entropy Solutions](https://mlsanity.com/paper/2406.05694/) (2024)

## Abstract (arXiv)

Classical reduced models are low-rank approximations using a fixed basis designed to achieve dimensionality reduction of large-scale systems. In this work, we introduce reduced deep networks, a generalization of classical reduced models formulated as deep neural networks. We prove depth separation results showing that reduced deep networks approximate solutions of parametrized hyperbolic partial differential equations with approximation error ε with O(|log(ε)|) degrees of freedom, even in the nonlinear setting where solutions exhibit shock waves. We also show that classical reduced models achieve exponentially worse approximation rates by establishing lower bounds on the relevant Kolmogorov N-widths.