# Approximation Rates for Metaplectic Neural Networks

arXiv: https://arxiv.org/abs/2608.08872 · HTML: https://mlsanity.com/paper/2608.08872/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 53 references that are in the map.
- First posted on arXiv 9 Aug 2026.

## Builds on (most cited first, 13 of 13)

- [Breaking the Curse of Dimensionality with Convex Neural Networks](https://mlsanity.com/paper/1412.8690/) (2014)
- [The Expressive Power of Neural Networks: A View from the Width](https://mlsanity.com/paper/1709.02540/) (2017)
- [Deep Neural Network Approximation Theory](https://mlsanity.com/paper/1901.02220/) (2019)
- [Sharp Bounds on the Approximation Rates, Metric Entropy, and n-widths of Shallow Neural Networks](https://mlsanity.com/paper/2101.12365/) (2021)
- [On the approximation of functions by tanh neural networks](https://mlsanity.com/paper/2104.08938/) (2021)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Characterization of the Variation Spaces Corresponding to Shallow Neural Networks](https://mlsanity.com/paper/2106.15002/) (2021)
- [Some observations on high-dimensional partial differential equations with Barron data](https://mlsanity.com/paper/2012.01484/) (2020)
- [Approximations with deep neural networks in Sobolev time-space](https://mlsanity.com/paper/2101.06115/) (2020)
- [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)
- [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)

## Abstract (arXiv)

In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform. Then, after establishing embedding between metaplectic Barron spaces and Sobolev spaces we consider a neural metaplectic dictionary and we prove Monte-Carlo approximation bounds for metaplectic Barron functions using finite linear combinations of atoms of the dictionary. Finally, we validate the introduction of the neural metaplectic dictionary by devising a deep neural network architecture that uses as building blocks the atoms of the dictionary. We test it to approximate solutions of time-dependent Schr"odinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.