基于函数值再生核希尔伯特空间的偏微分方程神经求解算子  

Neural Solution Operator of PDEs Based on Functionvalued Reproducing Kernel Hilbert Space

作　　者：包凯君 刘子源 王海峰 钱旭 宋松和[1] Bao Kaijun;Liu Ziyuan;Wang Haifeng;Qian Xu;Song Songhe(Department of Mathematics,National University of Defense Technology,Changsha 410073,China)

机构地区：[1]国防科技大学数学系,长沙410073

出　　处：《数学理论与应用》2023年第2期16-31,共16页Mathematical Theory and Applications

基　　金：supported by the National Key R&D Program of China(No.2020YFA0709800);the National Natural Science Foundation of China(Nos.11901577,11971481,12071481);the Natural Science Foundation of Hunan(Nos.2021JJ20053,2020JJ5652);the Science and Technology Innovation Program of Hunan Province(No.2021RC3082);the Defense Science Foundation of China(No.2021-JCJQ-JJ-0538)。

摘　　要：通过精心设计神经网络结构来学习无穷维函数空间之间的映射,算子学习方法——神经算子,相较于传统方法在求解偏微分方程等复杂问题上展现出极高的效率.为此,本文结合函数值再生核希尔伯特空间,提出一种新型的神经算子——再生核神经算子(RKNO).受到最近优秀的算子学习方法——深度算子网络(DeepONet)的启发,RKNO通过推广希尔伯特-施密特积分算子和表示定理而实现.在Advection,KdV,Burgers和Poisson方程上的数值实验表明,与DeepONet和其他模型相比,RKNO具有更易于表达和高效的结构.此外,RKNO还显示出与离散化无关的性质,可以在低分辨率数据训练后,找到高分辨率输入后的解.By learning the mappings between infinite dimensional function spaces using carefully designed neural networks,the operator learning methodology–neural operator has exhibited significantly more efficient than traditional methods in solving complex problems such as differential equations.Toward this end,we incorporate the functionvalued reproducing kernel Hilbert spaces(functionvalued RKHS)and propose a novel neural operator–reproducing kernel neural operator(RKNO).Motivated by the recently successful operator learning methodology–deep operator network(DeepONet),RKNO is formulated by generalizing the HilbertSchmidt integral operator and the representer theorem.Numerical experiments on the Advection,KdV,Burgers’,and Poisson equations show that the RKNO allows for an expressive and efficient architecture,in contrast to DeepONet and other models.Futhermore,the RKNO possesses the property of discretizationindependence,which can find the solution of a highresolution input after learning from lowresolution data.

关 键 词：神经网络 偏微分方程 函数型再生核希尔伯特空间 神经算子 深度算子网络 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程] O175[自动化与计算机技术—控制科学与工程]