# C1-approximation with rational functions and rational neural networks

arXiv: https://arxiv.org/abs/2508.19672 · HTML: https://mlsanity.com/paper/2508.19672/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 27 references that are in the map.
- First posted on arXiv 27 Aug 2025.

## Builds on (most cited first, 12 of 12)

- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [Extrapolation and learning equations](https://mlsanity.com/paper/1610.02995/) (2016)
- [Error bounds for approximations with deep ReLU neural networks in Ws,p norms](https://mlsanity.com/paper/1902.07896/) (2019)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Rational neural networks](https://mlsanity.com/paper/2004.01902/) (2020)
- [Approximation spaces of deep neural networks](https://mlsanity.com/paper/1905.01208/) (2019)
- [Neural networks and rational functions](https://mlsanity.com/paper/1706.03301/) (2017)
- [Simultaneous approximation of a smooth function and its derivatives by deep neural networks with piecewise-polynomial activations](https://mlsanity.com/paper/2206.09527/) (2022)
- [Semiglobal optimal feedback stabilization of autonomous systems via deep neural network approximation](https://mlsanity.com/paper/2002.08625/) (2020)
- [A descent algorithm for the optimal control of ReLU neural network informed PDEs based on approximate directional derivatives](https://mlsanity.com/paper/2210.07900/) (2022)
- [On uniqueness in structured model learning](https://mlsanity.com/paper/2410.22009/) (2024)
- [First-order conditions for the optimal control of learning-informed nonsmooth PDEs](https://mlsanity.com/paper/2206.00297/) (2022)

## Built on by (most cited first, 2 of 2)

- [Symbolic recovery of PDEs from measurement data](https://mlsanity.com/paper/2602.15603/) (2026)
- [All you need is SAMPAT](https://mlsanity.com/paper/2607.09235/) (2026)

## Abstract (arXiv)

We show that suitably regular functions can be approximated in the C1-norm both with rational functions and rational neural networks, including approximation rates with respect to width and depth of the network, and degree of the rational functions. As consequence of our results, we further obtain C1-approximation results for rational neural networks with the EQL^div and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.