发表机构
Johann Radon Institute of Computational and Applied Mathematics (RICAM); Austrian Academy of Sciences; University of Twente; Università degli Studi di Torino（约翰·拉东计算与应用数学研究所（RICAM）; 奥地利科学院; 特温特大学; 都灵大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究扩展巴伦空间概念并引入神经元坡涅特字典，证明其逼近界，构建对应深度网络架构，在含时薛定谔方程解的逼近任务中性能优于经典物理信息神经网络。
AI 中文摘要
本文针对基于元坡涅特算子字典构建的浅层神经网络，开展了定量逼近研究。首先，我们通过考虑傅里叶变换的辛动机扩展（即元坡涅特变换），扩展了巴伦空间的概念。在确立元坡涅特巴伦空间与索伯列夫空间之间的嵌入关系后，我们引入神经元坡涅特字典，并证明了利用该字典原子的有限线性组合对元坡涅特巴伦函数的蒙特卡洛逼近界。最后，我们设计了一种以该字典原子为构建模块的深度神经网络架构，验证了神经元坡涅特字典的引入效果。我们将其用于逼近含时薛定谔方程的解，与经典物理信息神经网络架构相比，展现出更优的性能。
英文摘要
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.
