# Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains

arXiv: https://arxiv.org/abs/2411.04108 · HTML: https://mlsanity.com/paper/2411.04108/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 46 references that are in the map.
- First posted on arXiv 6 Nov 2024.

## Builds on (most cited first, 17 of 17)

- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Understanding neural networks with reproducing kernel Banach spaces](https://mlsanity.com/paper/2109.09710/) (2021)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Ab-initio quantum chemistry with neural-network wavefunctions](https://mlsanity.com/paper/2208.12590/) (2022)
- [Ridges, Neural Networks, and the Radon Transform](https://mlsanity.com/paper/2203.02543/) (2022)
- [Approximation capabilities of neural networks on unbounded domains](https://mlsanity.com/paper/1910.09293/) (2019)
- [Weighted variation spaces and approximation by shallow ReLU networks](https://mlsanity.com/paper/2307.15772/) (2023)
- [Noncompact uniform universal approximation](https://mlsanity.com/paper/2308.03812/) (2023)
- [Lp sampling numbers for the Fourier-analytic Barron space](https://mlsanity.com/paper/2208.07605/) (2022)
- [Space-Time Approximation with Shallow Neural Networks in Fourier Lebesgue spaces](https://mlsanity.com/paper/2312.08461/) (2023)

## Built on by (most cited first, 5 of 5)

- [Any-Dimensional Invariant Universality](https://mlsanity.com/paper/2605.23156/) (2026)
- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [Time-Frequency Analysis for Neural Networks](https://mlsanity.com/paper/2512.15992/) (2025)
- [Approximation Capabilities of Feedforward Neural Networks with GELU Activations](https://mlsanity.com/paper/2512.21749/) (2025)
- [Approximation Rates for Metaplectic Neural Networks](https://mlsanity.com/paper/2608.08872/) (2026)

## Abstract (arXiv)

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.