基于残差神经网络模型的Fredholm积分方程数值解法  

Numerical method of Fredholm integral equation based on residual neural network model

作　　者：张殿焜 靳聪明[1] ZHANG Diankun;JIN Congming(School of Science,Zhejiang Sci-Tech University,Hangzhou 310018,China)

机构地区：[1]浙江理工大学理学院,杭州310018

出　　处：《浙江理工大学学报（自然科学版）》2020年第5期706-713,共8页Journal of Zhejiang Sci-Tech University(Natural Sciences)

基　　金：国家自然科学基金项目(11571314)。

摘　　要：为了求解Fredholm积分方程,特别是高维Fredholm积分方程,提出了一种采用残差神经网络求解Fredholm积分方程的数值方法。首先在求解区域随机产生训练数据集,通过前向传播残差神经网络得到训练集上的预测值;然后代入Fredholm积分方程得到离散格式,并定义损失函数,将解Fredholm积分方程转化为一个最小二乘问题;最后利用残差神经网络进行优化求解。该方法形式简单,对高维Fredholm积分方程求解问题计算量无显著增加。数值实验表明:该方法能有效求解Fredholm积分方程,且能取得很好的收敛精度;所训练的残差神经网络不会出现网络退化现象,表现出稳定性好、泛化能力强等优点。To solve Fredholm integral equations,especially high-dimensional Fredholm integral equations,a numerical method is proposed based on residual neural networks.Firstly,the training dataset was generated at random in the solution domain,and the predicted value on the training set was gained by forward propagation of residual neural network.Then,the discretization scheme was obtained by substituting the predicted value into the Fredholm integral equation,and the loss function was defined.Then,solving Fredholm integral equation was transformed into a least squares problem.Finally,residual neural network was used for optimizing the solution.The new method has a simple form and does not significantly increase the computational test for high-dimensional Fredholm integral equation problems.Numerical experimental results show that the new method can solve Fredholm integral equations efficiently and accurately and get good convergence precision.The residual neural networks will not suffer from network degradation,and has the advantages of good stability and strong generalization ability.

关 键 词：残差神经网络 FREDHOLM积分方程 高维积分方程 最小二乘法 

分 类 号：O175.5[理学—数学] TP183[理学—基础数学]