基于时间权重的物理信息神经网络求解非稳态偏微分方程  

SOLVING UNSTEADY PARTIAL DIFFERENTIAL EQUATIONS USING TIMEWEIGHTED PHYSICS-INFORMED NEURAL NETWORK

作　　者：韦昌 樊昱晨 周永清 刘欣[1,2] 李驰 王赫阳 Wei Chang;Fan Yuchen;Zhou Yongqing;Liu Xin;Li Chi;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》2025年第3期755-766,共12页Chinese Journal of Theoretical and Applied Mechanics

摘　　要：物理信息神经网络(physics-informed neural networks,PINN)是一种将深度学习技术与物理模型相结合的计算方法,目前已经成为智能科学计算领域的研究热点.然而,传统的PINN在求解非稳态偏微分方程时,常因忽视动态系统的时间因果关系而出现训练效率低下和预测精度不足等问题.为了解决这些局限性,文章提出了一种基于时间权重的物理信息神经网络(time-weighted physics-informed neural network,TWPINN).该模型通过引入可体现系统动态变化特征的时间权重函数,对损失函数进行了优化,以增强神经网络的时间因果关系.由于初始条件对于准确预测系统的动态行为具有决定性的作用,因此在TWPINN中,权重值会随着时间的推移而呈现单调递减的趋势,以确保模型对早期时间点采样数据赋予更高的重视.此外,TWPINN在训练过程中采用了一种动态权重调整策略,随着迭代次数的增加,模型会逐步调整对后续时间采样点的权重分配.这种策略使模型不仅能够捕捉到系统的初始状态和短期变化,而且显著提高了对长期演变趋势的预测准确性.为了验证TWPINN的性能,文章选取一维非稳态对流方程和一维非稳态反应扩散方程作为测试案例.结果表明,TWPINN在求解这两种方程时均能提供与基准解高度一致的预测结果,并且可以将预测误差控制在较低水平.Physics-informed neural networks(PINN)are advanced computational methods that integrate deep learning techniques with established physical models,and they have rapidly emerged as a research hotspot in the realm of intelligent scientific computing.However,despite their promising potential,traditional PINN often encounter significant challenges when applied to unsteady partial differential equations.The primary issue is their low training efficiency and insufficient prediction accuracy,which mainly stem from inadequate consideration of the temporal causality inherent in dynamic systems.To overcome those limitations,this paper introduces a novel approach known as the time-weighted physics-informed neural network,abbreviated as time-weighted physics-informed neural network(TWPINN).By introducing a well-designed time-weighted function that embodies the dynamic features of the system,the loss function is optimized to enhance the temporal causality in the neural network.Recognizing the pivotal role of initial conditions in accurately predicting the system’s long-term behavior,TWPINN assigns higher weights to the data points from earlier time stage.This is achieved through a monotonically decreasing weight value as time progresses,ensuring that the model pays greater attention to the initial state of the system.Additionally,TWPINN features a dynamic weight adjustment strategy during the training phase.As the number of training iterations increases,the model systematically adjusts the weight distribution for subsequent time samples.This dynamic strategy enables the model not only to capture the initial state and short-term variations of the system more effectively but also to significantly enhance its accuracy in predicting long-term evolutionary trends.To validate the performance of TWPINN,we selected the one-dimensional unsteady convection equation and the one-dimensional unsteady reaction-diffusion equation as test cases.The experimental results highlight that TWPINN can produce predictions that are highly consistent

关 键 词：物理信息神经网络 时间权重函数 对流方程 反应扩散方程 

分 类 号：TP183[自动化与计算机技术—控制理论与控制工程]