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

• URL: http://arxiv.org/abs/2508.19672v1
• Date: Wed, 27 Aug 2025 08:31:25 GMT
• Title: $\mathcal{C}^1$-approximation with rational functions and rational neural networks
• Abstract summary: We show that suitably regular functions can be approximated in the $mathcalC1$-norm both with rational functions and rational neural networks.<n>As consequence of our results, we obtain $mathcalC1$-approximation results for rational neural networks with the $EQLdiv$ and ParFam architecture.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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