Guan Yu, Fang Tingting, Zhang Diankun, Jin Congming
Department of Mathematics, Zhejiang Sci-Tech University, Hangzhou, 310018 China.
Int J Appl Comput Math. 2022;8(2):87. doi: 10.1007/s40819-022-01288-3. Epub 2022 Mar 29.
The aim of this paper is to provide a deep learning based method that can solve high-dimensional Fredholm integral equations. A deep residual neural network is constructed at a fixed number of collocation points selected randomly in the integration domain. The loss function of the deep residual neural network is defined as a linear least-square problem using the integral equation at the collocation points in the training set. The training iteration is done for the same set of parameters for different training sets. The numerical experiments show that the deep learning method is efficient with a moderate generalization error at all points. And the computational cost does not suffer from "curse of dimensionality" problem.
本文的目的是提供一种基于深度学习的方法，该方法能够求解高维弗雷德霍姆积分方程。在积分域中随机选择的固定数量的配置点处构建深度残差神经网络。深度残差神经网络的损失函数被定义为一个线性最小二乘问题，该问题使用训练集中配置点处的积分方程。针对不同训练集的同一组参数进行训练迭代。数值实验表明，深度学习方法效率高，在所有点处具有适度的泛化误差。并且计算成本不受“维数灾难”问题的影响。
