线性积分方程原函数变换深度学习求解方法  

Deep Learning Method for Solving Linear Integral Equations Through Primitive Function Transformation

作　　者：刘东[1,2,3] 陈奇隆 王雪强 Dong LIU;Qilong CHEN;Xueqiang WANG(Science and Technology on Reactor System Design Technology Laboratory,Nuclear Power Institute of China,Chengdu,Sichuan 610213,China;China National Nuclear Corporation Engineering Research Center of Nuclear Energy Software and Digital Reactor,Chengdu,Sichuan 610213,China;Science and Technology Committee of China National Nuclear Corporation,Beijing 100045,China)

机构地区：[1]中国核动力研究设计院核反应堆系统设计技术重点实验室,四川成都610213 [2]中国核工业集团核能软件与数字化反应堆工程技术研究中心,四川成都610213 [3]中国核工业集团有限公司科技委,北京100045

出　　处：《计算物理》2024年第5期651-662,共12页Chinese Journal of Computational Physics

基　　金：国家高层次人才特殊支持计划创新领军人才基金(J705981200002001)资助。

摘　　要：针对深度学习数值计算方法求解积分方程,提出求解线性积分方程的原函数变换深度学习方法,通过被积函数的原函数变换,将积分方程转化为纯粹的微分方程,并给出原函数定解条件确定方法,以及相应的神经网络损失函数生成方式。通过深度学习使得神经网络函数逼近原函数后,将原函数求导并根据积分核的形式进行逆变换,最终得到积分方程未知函数的数值解。通过多种典型算例数值实验证明,本文方法具有良好的精度与适用性,数值计算结果具有连续性的优点。Due to factors such as limited integral terms and approximations,solving integral equations using classical numerical methods is often more challenging than solving differential equations.This paper proposes a theory of solving linear integral equations through the transformation of primitive functions using deep learning.By transforming the integrand into a primitive function,the integral equation is converted into a purely differential equation.The paper also provides a method for determining the initial conditions of the primitive function and a technique for generating the neural network loss function.After approximating the primitive function using deep learning with neural networks,the derivative of the primitive function is calculated and transformed according to the form of the integral kernel,ultimately obtaining the numerical solution of the unknown function in the integral equation.Through numerical experiments on various typical examples,the paper demonstrates that the proposed theory and key techniques exhibit good accuracy and applicability,thereby opening up new technical approaches for the numerical solution of linear integral equations.

关 键 词：深度学习 线性积分方程 退化核 原函数变换 损失函数 数值验证 

分 类 号：TL329[核科学技术—核技术及应用]