$\mathcal{C}^1$-approximation with rational functions and rational neural networks
This work addresses the need for efficient symbolic regression in physical law learning, offering incremental improvements in approximation methods for specific architectures like EQL^÷ and ParFam.
The paper tackles the problem of approximating suitably regular functions in the C^1-norm using rational functions and rational neural networks, providing concrete approximation rates with respect to width, depth, and degree.
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 $\text{EQL}^÷$ and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.