证明了非多项式激活函数在非紧域上也能实现通用逼近。
Universal approximation results for neural networks with non-polynomial activation function over non-compact domains
- 用非多项式激活函数，在非紧域上实现通用逼近
- 覆盖 $L^p$、加权 $C^k$ 及加权 Sobolev 空间，含导数逼近
- 给出与维度无关的逼近速率，适用于傅里叶变换光滑可积函数
本文将单隐层前馈神经网络的通用逼近性质拓展至非紧域，特别关注在加权 $C^k$-空间和加权 Sobolev 空间中对无界域上的函数逼近。具体而言，在激活函数为非多项式的基础上，建立了在欧氏空间非紧子集上定义的函数空间中的通用逼近结果，包括 $L^p$-空间、加权 $C^k$-空间和加权 Sobolev 空间，后者支持对（弱）导数的逼近。此外，还给出了以具有足够正则性和可积傅里叶变换的函数为目标时，使用非多项式激活函数的神经网络所达到的维度无关逼近速率。
原文摘要 · Abstract (English)
This paper extends the universal approximation property of single-hidden-layer feedforward neural networks beyond compact domains, which is of particular interest for the approximation within weighted $C^k$-spaces and weighted Sobolev spaces over unbounded domains. More precisely, by assuming that the activation function is non-polynomial, we establish universal approximation results within function spaces defined over non-compact subsets of a Euclidean space, including $L^p$-spaces, weighted $C^k$-spaces, and weighted Sobolev spaces, where the latter two include the approximation of the (weak) derivatives. Moreover, we provide some dimension-independent rates for approximating a function with sufficiently regular and integrable Fourier transform by neural networks with non-polynomial activation function.
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