基于改进欧拉法的非线性偏微分方程神经网络求解器  

Neural network solver for nonlinear partial differential equations based on improved Euler method

作　　者：黄冠男 王靖岳 王美清[1] HUANG Guannan;WANG Jingyue;WANG Meiqing(College of Mathematics and Statistics,Fuzhou University,Fuzhou,Fujian 350108,China)

机构地区：[1]福州大学数学与统计学院,福建福州350108

出　　处：《福州大学学报（自然科学版）》2024年第4期396-403,共8页Journal of Fuzhou University(Natural Science Edition)

基　　金：国家自然科学基金资助项目(62172098)。

摘　　要：针对一般深度学习方法求解非线性偏微分方程时泛化能力差的问题,提出一种使用改进欧拉法联通网络模块的长短期卷积循环神经网络.该神经网络的构建运用改进欧拉法和有限差分法,通过改进欧拉法实现网络中模块之间的有效连接.基于有限差分法构建的卷积核实现偏微分方程中涉及的导数项的精确近似,并在Burgers方程和λ-ω反应扩散方程上进行仿真实验.实验结果证明,该方法不但在训练数据上具有很高的精度,而且在外推到新领域时也表现出较强的泛化能力.To address the poor generalization performance of conventional deep learning methods in solving nonlinear partial differential equations,a long short-term convolutional recurrent neural network with an improved Euler method connected network module is proposed.The construction of the neural network employs the improved Euler method and finite difference method.Effective connections between the modules are realized through the improved Euler method.The derivative terms involved in the partial differential equation are accurately approximated by convolutions kernels constructed based on the finite difference method.Simulation experiments are conducted on two typical nonlinear partial differential equations,namely the Burgers equation andλ-ωreaction-diffusion equation.The experi-mental results prove that this method not only has high precision on the training data,but also shows strong generalization ability when extrapolating to new fields.

关 键 词：偏微分方程 深度学习 长短期记忆 改进欧拉法 有限差分法 

分 类 号：O241.82[理学—计算数学] TP183[理学—数学]