基于PFNN-2的偏微分方程参数反演  

PARAMETER INVERSION OF PARTIAL DIFFERENTIAL EQUATION BASED ON PFNN-2

作　　者：张博[1,2] 盛海龙 杨超 Zhang Bo;Sheng Hailong;Yang Chao(National Engineering Laboratory for Big Data Analysis and Applications,Peking University,Beijing 100871,China;School of Mathematical Sciences,Peking University,Beijing 100871,China;Institute of Software,Chinese Academy of Sciences,Beijing 100190,China;PKU-Changsha Institute for Computing and Digital Economy,Changsha 410205,China)

机构地区：[1]北京大学大数据分析与应用技术国家工程实验室,北京100871 [2]北京大学数学科学学院,北京100871 [3]中国科学院软件研究所,北京100190 [4]北京大学长沙计算与数字经济研究院,长沙410205

出　　处：《数值计算与计算机应用》2024年第4期301-313,共13页Journal on Numerical Methods and Computer Applications

基　　金：国家自然科学基金(12131002,62306018);中国博士后科学基金(2022M710211)资助。

摘　　要：近年来,将人工神经网络用于求解偏微分方程正反问题的研究发展迅速,在正问题求解上,基于神经网络的Penalty-Free Neural Network-2(PFNN-2)方法可精确逼近问题的初始与本质边界条件,放松对解的光滑性要求,实现比较理想的求解精度(Sheng andYang,CiCP,2022)[1].在本文中,将结合PFNN-2的特点,将其扩展至偏微分方程参数反演问题当中.为了实现该目标,在原PFNN-2损失函数基础上,引入数据驱动损失项,同时制定了相应的平衡系数自适应策略.在数值实验中以Burgers方程及对流扩散方程中的参数反演为例,对提出的反演方法进行了测试,验证了方法的可行性.本研究扩展了PFNN-2方法的应用范围.In recent years,the researches on employing artificial neural networks to solve forward and inverse problems involving partial differential equations have developed rapidly.In solving the forward problems,Penalty-Free Neural Network-2(PFNN-2)method can accurately approximate the initial and essential boundary conditions of the problem,relax the smoothness requirement about the solution,and achieve satisfactory solution accuracy(Sheng and Yang,CiCP,2022)[1].In this paper,we extend PFNN-2 to the parameter inversion problem of partial differential equation by combining its characteristics.To achieve this goal,a data-driven loss term is introduced on the basis of the original PFNN-2 loss function,and an adaptive strategy for the corresponding balance coeficient is designed.In numerical experiments,taking inversions of parameters in Burgers equation and convection-diffusion equation as examples,the proposed inversion method is tested,validating its feasibility.This study extends the application scope of the PFNN-2 method.

关 键 词：偏微分方程 参数反演 神经网络 弱形式 数据驱动 PFNN-2 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程] O241.82[自动化与计算机技术—控制科学与工程]