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.
Code (0)
등록된 구현이 없습니다.
Similar Papers 제목 키워드 기반
Universal approximation results for neural networks with non-polynomial activation function over non-compact domains
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 we…
Time-Frequency Analysis for Neural Networks
We develop a quantitative approximation theory for shallow neural networks using tools from time-frequency analysis. Working in weighted modulation spaces $M^{p,q}_m(\mathbf{R}^{d})$, we prove dimension-independent appro…
Domain-Decomposed Randomized Neural Networks for Partial Differential Equations in Unbounded Domains
Partial differential equations on unbounded domains are challenging because the exterior region must be represented without excessive truncation error. Truncation-based methods often require problem-dependent artificial …
Integral Operator Approaches for Scattered Data Fitting on Spheres
This paper focuses on scattered data fitting problems on spheres. We study the approximation performance of a class of weighted spectral filter algorithms, including Tikhonov regularization, Landaweber iteration, spectra…
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 classe…