深度学习方法求解中子输运方程的微分变阶理论  

Differential Transform Order Theory for Solving Neutron TransportEquation by Deep Learning Method

作　　者：刘东[1,2,3] 王雪强 张斌 俞蔡阳[3] 宫兆虎 陈奇隆 LIU Dong;WANG Xueqiang;ZHANG Bin;YU Caiyang;GONG Zhaohu;CHEN Qilong(Nuclear Power Institute of China,Chengdu 610213,China;CNNC Engineering Research Center of Nuclear Energy Software and Digital Reactor,Chengdu 610213,China;College of Computer Science,Sichuan University,Chengdu 610065,China)

机构地区：[1]中国核动力研究设计院,四川成都610213 [2]中核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]四川大学计算机学院,四川成都610065

出　　处：《原子能科学技术》2023年第5期946-959,共14页Atomic Energy Science and Technology

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

摘　　要：中子输运方程是核反应堆物理分析计算的基本方程,针对深度学习技术求解输运方程因定积分项带来的困难,本文提出了微分变阶理论:将输运方程定积分项变换为对应的原函数,其他部分的角通量密度项表示为原函数高阶微分形式,从而将具有微积分形式的输运方程转换为纯粹的高阶微分方程;给出了变换后的原函数定解约束条件,以及对应的边界条件形式;构造了由原函数方程、边界条件、原函数定解、特征值约束共同形成的加权损失函数,利用深度学习使得神经网络逼近原函数;通过将原函数求导进行微分降阶,最终得到原输运方程角通量密度的数值解。针对多个平板、球几何例题进行了数值验证,获得了具有连续性特点的计算结果,证明了本文理论及相关方法的正确性,从而为中子输运方程的数值求解方法探索新的技术途径。The neutron transport equation is the primary equation for nuclear reactor physical analysis and calculation,which are usually solved by discrete ordinate(S N)method,method of characteristics(MOC),Monte Carlo method,and other numerical methods in tradition.In recent years,technical research on solving differential equations using deep learning methods has gradually become a cutting-edge hot topic in the field of numerical computation.Significant progress has been made in solving the differential equations.However,it is hard to solve the transport equation using deep learning methods directly.The main reason is that there are several multiple integral terms of angular flux density in the transport equation,which leads to the original method not being available.In order to address these challenges,the differential transform order theory of neutron transport equation was proposed in this work.The basic idea of this method is to transform the complex neutron transport equation with integral terms of angular flux density into a pure high-order differential equation.In the theory,the integral terms of the transport equation were transformed into their corresponding primitive function by mathematical method.Then,other terms of the neutron transport function were correspondingly expressed as a high-order differential form of the primitive function.Since the primitive function corresponding to definite integral is a multi-function family,the constraint conditions for determining the definite solution of the primitive function were provided.For different boundary constraints of the transport equation,the corresponding boundary condition forms represented by the primitive function were also proposed.The weighted loss function was constructed based on the constraints including the control equation,boundary conditions,constraints on the definite solution of the primitive function,and eigenvalue constraints.The deep learning method was used to make the artificial neural network approximate the primitive function.After the trai

关 键 词：深度学习 中子输运方程 原函数 微分变阶 损失函数 数值验证 

分 类 号：TL329[核科学技术—核技术及应用]