基于软约束优化求解偏微分方程的物理神经网络及应用  

Applications of Physics⁃Informed Neural Network Based on Soft Constraint Optimization in Solving Partial Differential Equations

作　　者：唐韫 陈硕[1] 贺冬冬 TANG Yun;CHEN Shuo;HE Dongdong(School of Aerospace Engineering and Applied Mechanics,Tongji University,Shanghai 200092,China;School of Science and Engineering,The Chinese University of Hong Kong,Shenzhen 518172,Guangdong,China)

机构地区：[1]同济大学航空航天与力学学院,上海200092 [2]香港中文大学(深圳)理工学院,广东深圳518172

出　　处：《力学季刊》2023年第4期782-792,共11页Chinese Quarterly of Mechanics

基　　金：国家自然科学基金(11872283)。

摘　　要：本文使用物理神经网络(Physics-Informed Neural Networks,PINN)对偏微分方程(Partial Differential Equation,PDE)进行了正向求解,该方法将控制方程和边界条件编码为损失函数,并使用自动微分机制和随机梯度下降算法来更新超参数,成功建立软约束优化下的PINN正向求解模型.针对稳态案例求解,文章展示了二维泊肃叶-库埃特,无黏Taylor-Green涡以及Re为100时的二维顶盖驱动方腔流计算结果,并比较了不同采样策略和采样点集大小对预测结果的影响.对于非稳态案例,文章提出了使用时间标签和“门函数”来强化模型表征时序因果的能力,成功在相同参数设置下求解了传统PINN失效的Allen-Cahn方程.本文为偏微分方程的求解提供了一种新思路,具有广泛的应用前景.In this paper,we use Physics-Informed Neural Networks(PINN)to forward solve the partial differential equations(PDEs).By incorporating the control equations along with the boundary conditions into the loss function,the hyperparameters are updated by the automatic differentiation mechanism and the stochastic gradient descent algorithm,and the PINN forward solving model based on soft constraint optimization is successfully established.With respect to the steady state case,the performance of the PINN framework is demonstrated through several examples such as the two-dimensional Poiseuille-Couette,the non-viscous Taylor-Green vortices and the lid-driven cavity flow with Re=100,and the effects of different sampling strategies and sampling point set sizes on the predicted results are compared.On the other hand,for the unsteady state case,we provide guidelines for enhancing the model's ability to represent temporal causality by using time labels and"gate functions".We have demonstrated the capability of the proposed method through successfully solving the Allen-Cahn equation for which traditional PINN formulations fail to solve under the same parameter settings.This method provides a new approach to solve the partial differential equations and is believed to possess promising application potential.

关 键 词：PINN 正演 稳态求解 非稳态求解 时序因果 

分 类 号：O35[理学—流体力学]