基于PINN的Burgers方程求解模型  

Solution Model of Burgers Equation Based on PINN

作　　者：骆炜杰 李芳 陈鑫 LUO Weijie;LI Fang;CHEN Xin(Information Engineering University,Zhengzhou 450001,China;National Research Center of Parallel Computer Engineering and Technology,Beijing 100190,China)

机构地区：[1]信息工程大学,河南郑州450001 [2]国家并行计算机工程技术研究中心,北京100190

出　　处：《信息工程大学学报》2024年第3期323-330,共8页Journal of Information Engineering University

基　　金：国家重点研发计划(2020YFB0204800)。

摘　　要：传统的数值求解方法面临维数灾难和效率与精度平衡问题,而基于数据驱动的神经网络求解方法又存在训练量冗余和不可解释性问题。针对此问题,物理信息神经网络(Physical Information Neural Networks,PINNs)关注了训练数据中隐含的物理先验知识,融合了神经网络拟合复杂变量的能力,赋予了传统神经网络所缺乏的物理可解释性。应用该算法模型,提出了一种基于PINN的Burgers方程求解模型,该算法模型在训练中施加物理信息约束,因此能用少量的训练样本学习预测到分布在时空域上的偏微分方程模型。实验结果表明,在1+1维Burgers方程算例下,所提方法相比于经典的机器学习算法能有效捕抓到方程的变化并进行精确模拟,相比于有限差分法,可以大幅度缩短模拟时间。通过对不同的网络参数进行比较实验,所提方法在10%的噪声破坏下能产生合理的识别准确度,网络逼近方程的待定系数误差在0.001以内。The traditional numerical solution methods face the problems of dimension disaster and efficiency and accuracy balance,while the neural network solution method based on data-driven has the problems of training redundancy and inexplicability.To solve this problem,physical information neural networks(PINNs)pay attention to the physical prior knowledge implied in the training data,integrate the ability of neural networks to fit complex variables,and endow the traditional neural networks with the physical interpretability that is lacking.By applying the algorithm model,a solution model of Burgers equation based on PINN is proposed.The algorithm model imposes physical information constraints during training,so it can use a small number of training samples to learn and predict the partial differential equation model distributed in the space-time domain.The experimental results show that in the case of 1+1 dimensional Burgers equation,compared with the classical machine learning algorithm,the proposed method can effectively catch the changes of the equation and simulate accurately,and can significantly shorten the simulation time compared with the finite difference method.Through comparative experiments on different network parameters,even under 10%noise damage,it can produce reasonable recognition accuracy,and the undetermined coefficient error of network approximation equation is within 0.001.

关 键 词：计算流体力学 深度学习 物理信息神经网络 BURGERS方程 

分 类 号：TP181[自动化与计算机技术—控制理论与控制工程]