用有理函数和有理神经网络实现光滑函数的精确逼近，支持物理定律学习。
$\mathcal{C}^1$-approximation with rational functions and rational neural networks
- 基于有理函数与有理网络结构，实现$ C^1$范数下的高精度逼近
- 给出网络宽度、深度与有理函数阶数对逼近误差的影响规律
- 适用于符号回归中的物理定律发现，尤其适配EQL^÷与ParFam架构
我们证明了适当正则的函数可以用有理函数和有理神经网络在$ C^1$-范数下进行逼近，并给出了关于网络宽度、深度以及有理函数阶数的逼近速率。由此结果进一步得到，具有EQL^÷和ParFam架构的有理神经网络也具备$ C^1$-逼近能力，这两类架构在物理定律学习的符号回归中具有重要意义。
原文摘要 · Abstract (English)
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.
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