基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解  

Solving forward and inverse problems of the heat conduction equation using physics-informed neural networks

作　　者：赵暾 周宇[1,2] 程艳青 钱炜祺[1,2] ZHAO Tun;ZHOU Yu;CHENG Yanqing;QIAN Weiqi(State Key Laboratory of Aerodynamics,Mianyang 621000,China;China Aerodynamics Research and Development Center,Mianyang 621000,China)

机构地区：[1]空气动力学国家重点实验室,绵阳621000 [2]中国空气动力研究与发展中心,绵阳621000

出　　处：《空气动力学学报》2021年第5期19-26,共8页Acta Aerodynamica Sinica

摘　　要：建立了一种基于内嵌物理机理神经网络(PINN)的热传导方程的正问题及逆问题求解方法。该方法利用自动微分技术将一维热传导方程嵌入到深度网络的损失函数中,通过以损失函数最小为目标来优化深度网络,求解一维热传导方程以及对方程中的未知导热系数进行辨识。随后,分析了基于PINN求解正问题的收敛精度以及参数辨识的鲁棒性,并得出以下结论:在给定网络结构的情况下,基于PINN求解一维热传导方程的收敛误差在样本点数较少时主要由采样误差主导,而当样本点数较多时,收敛误差由优化误差主导;由于损失函数中包含了方程相关的正则化项,以及采用了自动微分技术,因此,基于PINN的参数辨识方法噪声标签数据具有较强的鲁棒性。A method for solving the forward and inverse problems of the heat conduction equation based on a physics-informed neural network(PINN)is proposed.In this method,the one-dimensional heat conduction equation is integrated into a loss function as the regularization term using the automatic differentiation.Consequently,the solution of the equation as well as the embedded unknown physical parameters can both be obtained by minimizing the loss function of the PINN.The convergence errors of solving the forward problem and the robustness of parameter identification are analyzed.Results show that for a given network structure,the convergence errors of solving the one-dimensional heat conduction equation based on PINN are is mainly determined by the sampling errors when sampling points are insufficient,while for abundant samples,the convergence error is determined by the optimization errors.Since the loss function contains equation-related regularization terms and adopts the automatic differentiation technique,the identification method based on PINN is robust to noisy data.

关 键 词：基于内嵌物理机理神经网络 热传导方程 正问题及逆问题 收敛精度 鲁棒性 

分 类 号：V231.13[航空宇航科学与技术—航空宇航推进理论与工程]