Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains

TL;DR

This paper extends the understanding of shallow neural network approximation capabilities in weighted Sobolev spaces, including unbounded domains and decaying weights, providing new embedding results and asymptotic approximation rates without curse of dimensionality.

Contribution

It introduces new approximation rates for neural networks on unbounded domains with weighted Sobolev spaces, expanding prior results to more general settings.

Findings

Established embedding results for weighted Fourier-Lebesgue spaces

Derived asymptotic approximation rates without curse of dimensionality

Extended approximation theory to unbounded domains with decaying weights

Abstract

In this work, we consider the approximation capabilities of shallow neural networks in weighted Sobolev spaces for functions in the spectral Barron space. The existing literature already covers several cases, in which the spectral Barron space can be approximated well, i.e., without curse of dimensionality, by shallow networks and several different classes of activation function. The limitations of the existing results are mostly on the error measures that were considered, in which the results are restricted to Sobolev spaces over a bounded domain. We will here treat two cases that extend upon the existing results. Namely, we treat the case with bounded domain and Muckenhoupt weights and the case, where the domain is allowed to be unbounded and the weights are required to decay. We first present embedding results for the more general weighted Fourier-Lebesgue spaces in the weighted Sobolev spaces and then we establish asymptotic approximation rates for shallow neural networks that come without curse of dimensionality.