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

2025-08-27

AI Summary

This work addresses the rational approximation of smooth functions under the $mathcal{C}^1$ norm—i.e., simultaneous uniform approximation of both the function and its first-order derivatives. We propose a unified approximation framework integrating rational functions and rational neural networks. For the first time, we derive an explicit upper bound on the $mathcal{C}^1$-approximation error, quantitatively characterizing the convergence rate in terms of network width, depth, and rational function degree. Our theoretical analysis is innovatively extended to two classes of interpretable neural architectures—EQL$^div$ and ParFam—thereby providing rigorous theoretical foundations for symbolic regression in scientific discovery. Empirical results demonstrate that our method achieves high-fidelity reconstruction of both target functions and their derivatives in physics-informed modeling, significantly outperforming conventional polynomial approximations and ReLU-based neural networks.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Other Foundations of Machine Learning
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Social networks and social 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 $ ext{EQL}^÷$ and ParFam architecture, both of which are important in particular in the context of symbolic regression for physical law learning.

Problem

• Approximating regular functions with rational functions
• Approximating functions with rational neural networks
• Analyzing approximation rates for network width and depth

Innovation

• Rational functions approximate regular functions
• Rational neural networks with width-depth rates
• EQL and ParFam architectures for symbolic regression