Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains

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.

Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces

Approximation capabilities of shallow neural networks (SNNs) form an integral part in understanding the properties of deep neural networks (DNNs). In the study of these approximation capabilities some very popular classes of target functions are the so-called spectral Barron spaces. This spaces are of special interest when it comes to the approximation of partial differential equation (PDE) solutions. It has been shown that the solution of certain static PDEs will lie in some spectral Barron space. In order to alleviate the limitation to static PDEs and include a time-domain that might have a different regularity than the space domain, we extend the notion of spectral Barron spaces to anisotropic weighted Fourier-Lebesgue spaces. In doing so, we consider target functions that have two blocks of variables, among which each block is allowed to have different decay and integrability properties. For these target functions we first study the inclusion of anisotropic weighted Fourier-Lebesgue spaces in the Bochner-Sobolev spaces. With that we can now also measure the approximation error in terms of an anisotropic Sobolev norm, namely the Bochner-Sobolev norm. We use this observation in a second step where we establish a bound on the approximation rate for functions from the anisotropic weighted Fourier-Lebesgue spaces and approximation via SNNs in the Bochner-Sobolev norm.