We investigate the approximation of functions $f$ on a bounded domain $\Omega\subset \mathbb{R}^d$ by the outputs of single-hidden-layer ReLU neural networks of width $n$. This form of nonlinear $n$-term dictionary approximation has been intensely studied since it is the simplest case of neural network approximation (NNA). There are several celebrated approximation results for this form of NNA that introduce novel model classes of functions on $\Omega$ whose approximation rates avoid the curse of dimensionality. These novel classes include Barron classes, and classes based on sparsity or variation such as the Radon-domain BV classes. The present paper is concerned with the definition of these novel model classes on domains $\Omega$. The current definition of these model classes does not depend on the domain $\Omega$. A new and more proper definition of model classes on domains is given by introducing the concept of weighted variation spaces. These new model classes are intrinsic to the domain itself. The importance of these new model classes is that they are strictly larger than the classical (domain-independent) classes. Yet, it is shown that they maintain the same NNA rates.

翻译：本文研究有界域 $\Omega\subset \mathbb{R}^d$ 上函数 $f$ 通过宽度为 $n$ 的单隐层ReLU神经网络输出逼近的问题。这种非线性 $n$ 项字典逼近形式因是神经网络逼近(NNA)的最简情形而受到广泛研究。针对此类NNA逼近，已有若干经典结果引入能避免维数灾难的新型函数模型类，包括Barron类、基于稀疏性或变差的Radon域BV类等。本文旨在探讨域 $\Omega$ 上这些新型模型类的定义问题。当前模型类的定义不依赖于域 $\Omega$，本文通过引入加权变差空间的概念，给出更合适的域上模型类新定义。这些新型模型类具有域内在特性，其重要性在于虽严格大于经典（域无关）模型类，但仍能保持相同的NNA逼近速率。