Approximation Rates for Metaplectic Neural Networks

2026-08-09

AI Summary

This work addresses the challenge that conventional neural networks struggle to effectively approximate solutions to problems with symplectic structures or quantum dynamics, such as the time-dependent Schrödinger equation. It introduces the metaplectic transform into neural network theory for the first time, constructing a neural dictionary based on this transform and defining a corresponding metaplectic Barron space. The study establishes embedding relations between this space and Sobolev spaces, providing a theoretical foundation for a novel deep network architecture. This architecture leverages finite linear combinations to achieve Monte Carlo approximation of metaplectic Barron functions. Numerical experiments demonstrate that the proposed method significantly outperforms classical physics-informed neural networks in solving the time-dependent Schrödinger equation, thereby validating the expressive power and effectiveness of the introduced dictionary.

Abstract

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 Schrödinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.

Problem

• approximation rates
• metaplectic neural networks
• Schrödinger equations
• Barron spaces
• neural network approximation

Innovation

• metaplectic transform
• Barron space
• neural dictionary
• Monte-Carlo approximation
• physics-informed neural networks