基于有限差分残差物理约束的波动方程无监督学习方法  

Unsupervised learning method for the wave equation based on finite difference residual constraints loss

作　　者：冯鑫 姜屹[3] 秦嘉贤 张来平 邓小刚 FENG Xin;JIANG Yi;QIN Jia-Xian;ZHANG Lai-Ping;DENG Xiao-Gang(College of Computer Science,Sichuan University,Chengdu 610065,China;Tianfu Engineering-oriented Numerical Simulation&Software Innovation Center,Sichuan University,Chengdu 610207,China;Institute of Systems Engineering,Academy of Military Sciences,Beijing 100082,China;Institute of Defense Science and Technology Innovation,Academy of Military Sciences,Beijing 100071,China)

机构地区：[1]四川大学计算机学院,成都610065 [2]四川大学天府工程数值模拟与软件创新中心,成都610207 [3]军事科学院系统工程研究院,北京100082 [4]军事科学院国防科技创新研究院,北京100071

出　　处：《四川大学学报（自然科学版）》2024年第5期69-79,共11页Journal of Sichuan University(Natural Science Edition)

基　　金：国家重大专项(GJXM92579);基础科研计划(JCKY2022110C119)。

摘　　要：波动方程是一种重要的物理偏微分方程,近年来深度学习有望加速或替代传统数值方法对其求解.然而现有深度学习方法存在数据集获取成本高、训练效率低、边界条件泛化能力不足的问题,为此本文提出一种基于有限差分残差约束的波动方程无监督学习方法,基于结构网格和有限差分方法构建一种新颖的有限差分残差约束,以及一种无监督训练策略,使得卷积神经网络能够在无数据条件下训练,并预测波的正演过程.实验结果表明,有限差分残差约束相较于PINNs类的物理信息约束具有更容易拟合、计算成本更低、源项泛化能力更强的优点,这使得我们的方法有着更高的训练效率和应用潜力.The wave equation is an important physical partial differential equation,and in recent years,deep learning has shown potential to accelerate or replace traditional numerical methods for solving it.However,existing deep learning methods suffer from high data acquisition costs,low training efficiency,and insufficient generalization capability for boundary conditions.To address these issues,this paper proposes an unsupervised learning method for the wave equation based on finite difference residual constraints.The authors construct a novel finite difference residual constraint based on structured grids and finite difference methods,as well as an unsupervised training strategy,enabling convolutional neural networks to train without data and predict the forward propagation process of waves.Experimental results demonstrate that finite difference residual constraints have advantages over Physics-Informed Neural Networks(PINNs)type physical information constraints,such as easier fitting,lower computational costs,and stronger source term generalization capability,making our method more efficient in training and potent in application.

关 键 词：卷积神经网络 有限差分方法 波动方程 无监督学习 

分 类 号：TP391[自动化与计算机技术—计算机应用技术]