用旋转变换构造神经网络，逼近量子方程解更高效。
Approximation Rates for Metaplectic Neural Networks
- 基于旋转变换构建神经网络字典，扩展巴龙函数空间
- 证明有限线性组合可逼近元函数，误差随样本数下降
- 在含时薛定谔方程求解中优于传统物理信息网络
本文针对基于旋转变换（metaplectic operators）构建的浅层神经网络，建立定量逼近理论。首先，通过引入受辛几何启发的元傅里叶变换——元变换（metaplectic transform），拓展巴龙空间概念。随后，建立了元巴龙空间与 Sobolev 空间之间的嵌入关系，并分析了由元变换原子构成的神经网络字典对元巴龙函数的蒙特卡洛逼近性能，证明其逼近误差随参数数量增长而收敛。最后，设计一种以该字典原子为基本单元的深层网络架构，在含时薛定谔方程求解任务中验证其有效性，表现优于经典物理信息神经网络（PINNs）。实验在多个测试场景中展示出更优的精度与泛化能力。
原文摘要 · Abstract (English)
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.
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