Approximation Rates for Metaplectic Neural Networks
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Computer Science > Machine Learning
Title:Approximation Rates for Metaplectic Neural Networks
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.
| Subjects: | Machine Learning (cs.LG); Functional Analysis (math.FA) |
| Cite as: | arXiv:2608.08872 [cs.LG] |
| (or arXiv:2608.08872v1 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2608.08872
arXiv-issued DOI via DataCite (pending registration)
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