基于Runge-Kutta的自回归物理信息神经网络求解偏微分方程  

SELF-REGRESSIVE PHYSICS-INFORMED NEURAL NETWORK BASED ON RUNGE-KUTTA METHOD FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS

作　　者：韦昌 樊昱晨 周永清 张超群 刘欣[1,2] 王赫阳 Wei Chang;Fan Yuchen;Zhou Yongqing;Zhang Chaoqun;Liu Xin;Wang Heyang(School of Mechanical Engineering,Tianjin University,Tianjin 300072,China;Yantai Longyuan Power Technology Co.,Ltd.,Yantai 264006,Shandong,China)

机构地区：[1]天津大学机械工程学院,天津300072 [2]烟台龙源电力技术股份有限公司,山东烟台264006

出　　处：《力学学报》2024年第8期2482-2493,共12页Chinese Journal of Theoretical and Applied Mechanics

摘　　要：物理信息神经网络离散时间模型(PINN-RK)是深度学习技术与龙格库塔方法相结合的产物,在求解偏微分方程时具有非常出色的稳定性和较高的求解精度.但是,受到龙格库塔算法本身的限制,PINN-RK模型仅能实现单步时间预测,且计算效率较低.因此,为了实现多时间步长预测和提高模型的计算效率,提出了一种基于龙格库塔法的自回归物理信息神经网络模型(SR-PINN-RK).该模型基于自回归时间步进机制,改进了神经网络的训练流程和网络结构,相比PINN-RK模型,大幅减少了神经网络的训练参数,提高了模型的计算效率.此外,在自回归机制的作用下,该模型通过对标签数据的动态更新,成功实现了对偏微分方程解的多时间步长预测.为了验证文中模型的求解精度和计算效率,分别求解了Allen-Cahn方程和Burgers方程,并与文献中的基准解进行了对比.结果表明,模型预测解与基准解之间具有很高的一致性,求解Allen-Cahn方程和Burgers方程的最大相对误差均低于0.009.Physics-informed neural networks discrete-time model(PINN-RK)is a product of combining deep learning techniques with Runge-Kutta method,which has excellent stability and high accuracy in solving partial differential equations.As an emerging computational tool,PINN-RK has been widely applied in solving various complex problems in scientific and engineering fields.However,due to the limitations of the Runge-Kutta algorithm itself,the PINN-RK model can only achieve single-step time prediction and has low computational efficiency.To achieve multi-step time prediction and improve the computational efficiency of the model,this paper proposes a novel self-regressive physicsinformed neural networks model based on the Runge-Kutta method(SR-PINN-RK).The SR-PINN-RK model builds upon the PINN-RK model by incorporating a self-regressive time-advancing mechanism which allows the SR-PINN-RK model to learn the temporal dynamics of the partial differential equation more effectively,resulting in improved training performance and accuracy.In SR-PINN-RK model,except for the label data at the initial time given by the user,all other training labels are provided by the neural network model itself.The new PINN-RK model is a significant improvement over the PINN-RK model,with a much smaller number of training parameters and a significant boost in computational efficiency.This makes SR-PINN-RK model much faster and easier to train,while still maintaining the same level of accuracy.The SR-PINN-RK model uses a self-regressive mechanism to dynamically update the label data,which allows it to successfully achieve multi-step time prediction of partial differential equation solutions.This represents a remarkable improvement compared to the PINN-RK model,which is limited to single-step predictions.In order to verify the accuracy and computational efficiency of the SR-PINN-RK model,the Allen-Cahn equation and Burgers equation are solved using the SR-PINN-RK model and the predicted results are compared with the benchmark solutions in the literatu

关 键 词：物理信息神经网络 自回归时间步进机制 偏微分方程 Allen-Cahn方程 BURGERS方程 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程]