$\mathcal{C}^1$-approximation with rational functions and rational neural networks

TL;DR

This paper demonstrates that regular functions can be effectively approximated in the $ ext{C}^1$-norm using rational functions and neural networks, providing approximation rates related to network size and function degree.

Contribution

It introduces new $ ext{C}^1$-approximation results for rational functions and neural networks, including specific architectures relevant to symbolic regression and physical law discovery.

Findings

Approximation rates depend on network width, depth, and rational function degree.

Rational neural networks with specific architectures can achieve $ ext{C}^1$-approximation.

Results are relevant for symbolic regression and physical law learning.

Abstract

We show that suitably regular functions can be approximated in the $\mathcal{C}^1$-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 $\mathcal{C}^1$-approximation results for rational neural networks with the $\text{EQL}^\div$ and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.