摘要
为了探索基于物理信息的神经网络(PINN)求解微分方程时,物理信息在训练神经网络中的作用,提出将物理信息分为规律信息和数值信息2类,以阐释PINN求解微分方程的逻辑,以及物理信息的数据驱动方式和神经网络可解释性.设计基于2类信息的神经网络综合损失函数,并从训练采样和训练强度2方面建立信息的训练平衡度,从而利用PINN求解Burgers-Fisher方程.实验表明,PINN能够获得较好的方程求解精度和稳定性;在求解方程的神经网络训练中,Burgers-Fisher方程的数值信息比规律信息能更好地促进神经网络逼近方程解;随着训练采样和迭代次数的增加,以及2类信息的平衡,神经网络训练效果得到提高;增加神经网络规模可以提高方程求解精度,但也增加了网络训练迭代时间,在固定训练时间下并非神经网络规模越大效果越好.
Physical information was divided into rule information and numerical information,in order to explore the role of physical information in training neural network when solving differential equations with physics-informed neural network(PINN).The logic of PINN for solving differential equations was explained,as well as the datadriven approach of physical information and neural network interpretability.Synthetic loss function of neural network was designed based on the two types of information,and the training balance degree was established from the aspects of training sampling and training intensity.The experiment of solving the Burgers-Fisher equation by PINN showed that PINN can obtain good solution accuracy and stability.In the training of neural networks for solving the equation,numerical information of the Burgers-Fisher equation can better promote neural network to approximate the equation solution than rule information.The training effect of neural network was improved with the increase of training sampling,training epoch,and the balance between the two types of information.In addition,the solving accuracy of the equation was improved with the increasing of the scale of neural network,but the training time of each epoch was also increased.In a fixed training time,it is not true that the larger scale of the neural network,the better the effect.
作者
徐健
朱海龙
朱江乐
李春忠
XU Jian;ZHU Hai-long;ZHU Jiang-le;LI Chun-zhong(School of Statistics and Applied Mathematics,Anhui University of Finance and Economics,Bengbu 233030,China)
出处
《浙江大学学报（工学版）》
EI
CAS
CSCD
北大核心
2023年第11期2160-2169,共10页
Journal of Zhejiang University：Engineering Science
基金
国家自然科学基金资助项目(72131006,71971001,71803001)
安徽省教育厅高校自然科学研究重点资助项目(KJ2021A0473,KJ2021A0481,2022AH050608).
