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

arXiv: https://arxiv.org/abs/2312.08461 · HTML: https://mlsanity.com/paper/2312.08461/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 57 references that are in the map.
- First posted on arXiv 13 Dec 2023.

## Builds on (most cited first, 18 of 18)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [A Function Space View of Bounded Norm Infinite Width ReLU Nets: The Multivariate Case](https://mlsanity.com/paper/1910.01635/) (2019)
- [How do infinite width bounded norm networks look in function space?](https://mlsanity.com/paper/1902.05040/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [Approximation by Combinations of ReLU and Squared ReLU Ridge Functions with  ℓ1  and  ℓ0  Controls](https://mlsanity.com/paper/1607.07819/) (2016)
- [Representation formulas and pointwise properties for Barron functions](https://mlsanity.com/paper/2006.05982/) (2020)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Random feature neural networks learn Black-Scholes type PDEs without curse of dimensionality](https://mlsanity.com/paper/2106.08900/) (2021)
- [Neural network approximation and estimation of classifiers with classification boundary in a Barron class](https://mlsanity.com/paper/2011.09363/) (2020)
- [Risk Bounds for High-dimensional Ridge Function Combinations Including Neural Networks](https://mlsanity.com/paper/1607.01434/) (2016)
- [On the Representation of Solutions to Elliptic PDEs in Barron Spaces](https://mlsanity.com/paper/2106.07539/) (2021)
- [Complexity Measures for Neural Networks with General Activation Functions Using Path-based Norms](https://mlsanity.com/paper/2009.06132/) (2020)
- [A Priori Generalization Error Analysis of Two-Layer Neural Networks for Solving High Dimensional Schr"odinger Eigenvalue Problems](https://mlsanity.com/paper/2105.01228/) (2021)
- [Neural Network Approximations of PDEs Beyond Linearity: A Representational Perspective](https://mlsanity.com/paper/2210.12101/) (2022)
- [Lp sampling numbers for the Fourier-analytic Barron space](https://mlsanity.com/paper/2208.07605/) (2022)
- [Integral representations of shallow neural network with Rectified Power Unit activation function](https://mlsanity.com/paper/2112.11157/) (2021)
- [Bochner integrals and neural networks](https://mlsanity.com/paper/2302.13228/) (2023)

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

- [Weighted Sobolev Approximation Rates for Neural Networks on Unbounded Domains](https://mlsanity.com/paper/2411.04108/) (2024)
- [Time-Frequency Analysis for Neural Networks](https://mlsanity.com/paper/2512.15992/) (2025)
- [Sampling Complexity of Deep Approximation Spaces](https://mlsanity.com/paper/2312.13379/) (2023)
- [Fractional Parabolic Partial Differential Equations in Anisotropic Spectral Barron Spaces: Regularity and Neural Approximation](https://mlsanity.com/paper/2607.27781/) (2026)
- [Approximation Rates for Metaplectic Neural Networks](https://mlsanity.com/paper/2608.08872/) (2026)

## Abstract (arXiv)

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.