抛物型方程逆时反问题数据驱动解的泛化误差估计  

Estimation of Generalization Error for Data-Driven Solutions to Backward Problems of Parabolic Equations

作　　者：陈振兴 阮周生 万广红 周洁 李晓瑞 CHEN Zhenxing;RUAN Zhousheng;WAN Guanghong;ZHOU Jie;LI Xiaorui(East China University of Technology,Nanchang 330013,China)

机构地区：[1]东华理工大学理学院,江西南昌330013

出　　处：《昆明理工大学学报(自然科学版)》2025年第1期214-221,共8页Journal of Kunming University of Science and Technology(Natural Science)

基　　金：国家自然科学基金项目(12061008,11861007);江西省自然科学基金项目(20202BABL201004).

摘　　要：研究神经网络逼近抛物型方程逆时反问题解的收敛性问题.利用物理信息神经网络(Physics-Informed Neural Network,PINN)将抛物型方程逆时反问题转化为优化神经网络损失函数的网络参数问题,借助逆时反问题的条件稳定性结论及数值积分规则,建立了基于数据驱动的神经网络解与反问题精确解的泛化误差估计.通过数值实验验证了基于物理信息神经网络求解逆时反问题的有效性.This paper studies the convergence between the neural network approximation and the solution of the backward problem for the parabolic equation.The backward problem of parabolic equation is transformed into a parameter optimization problem for the loss function of neural networks by using the the Physics-Informed Neural Network(PINN).With the help of the conditional stability of the backward problem and numerical integration rule,the generalization error estimate between the data-driven neural network solution and the exact solution of inverse problem is established.The effectiveness of using physical information neural network to solve backward problem has been verified through numerical experiments.

关 键 词：物理信息神经网络 逆时问题 条件稳定性 泛化误差 

分 类 号：O175.26[理学—数学] TP183[理学—基础数学]