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Journal of ZheJiang University (Engineering Science)  2023, Vol. 57 Issue (11): 2160-2169    DOI: 10.3785/j.issn.1008-973X.2023.11.003
    
Solution approach of Burgers-Fisher equation based on physics-informed neural networks
Jian XU(),Hai-long ZHU*(),Jiang-le ZHU,Chun-zhong LI
School of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China
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Abstract  

Physical information was divided into rule information and numerical information, in order to explore the role of physical information in training neural network when solving differential equations with physics-informed neural network (PINN). The logic of PINN for solving differential equations was explained, as well as the data-driven approach of physical information and neural network interpretability. Synthetic loss function of neural network was designed based on the two types of information, and the training balance degree was established from the aspects of training sampling and training intensity. The experiment of solving the Burgers-Fisher equation by PINN showed that PINN can obtain good solution accuracy and stability. In the training of neural networks for solving the equation, numerical information of the Burgers-Fisher equation can better promote neural network to approximate the equation solution than rule information. The training effect of neural network was improved with the increase of training sampling, training epoch, and the balance between the two types of information. In addition, the solving accuracy of the equation was improved with the increasing of the scale of neural network, but the training time of each epoch was also increased. In a fixed training time, it is not true that the larger scale of the neural network, the better the effect.



Key wordsBurgers-Fisher equation      physics-informed neural network      regularity information      numerical information      data-driven      interpretability      training balance     
Received: 31 January 2023      Published: 11 December 2023
CLC:  TP 3  
Fund:  国家自然科学基金资助项目(72131006,71971001,71803001);安徽省教育厅高校自然科学研究重点资助项目(KJ2021A0473,KJ2021A0481,2022AH050608)
Corresponding Authors: Hai-long ZHU     E-mail: jianx1982@vip.163.com;hai-long-zhu@163.com
Cite this article:

Jian XU,Hai-long ZHU,Jiang-le ZHU,Chun-zhong LI. Solution approach of Burgers-Fisher equation based on physics-informed neural networks. Journal of ZheJiang University (Engineering Science), 2023, 57(11): 2160-2169.

URL:

https://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2023.11.003     OR     https://www.zjujournals.com/eng/Y2023/V57/I11/2160


基于物理信息神经网络的Burgers-Fisher方程求解方法

为了探索基于物理信息的神经网络(PINN)求解微分方程时,物理信息在训练神经网络中的作用,提出将物理信息分为规律信息和数值信息2类,以阐释PINN求解微分方程的逻辑,以及物理信息的数据驱动方式和神经网络可解释性.设计基于2类信息的神经网络综合损失函数,并从训练采样和训练强度2方面建立信息的训练平衡度,从而利用PINN求解Burgers-Fisher方程. 实验表明,PINN能够获得较好的方程求解精度和稳定性;在求解方程的神经网络训练中,Burgers-Fisher方程的数值信息比规律信息能更好地促进神经网络逼近方程解;随着训练采样和迭代次数的增加,以及2类信息的平衡,神经网络训练效果得到提高;增加神经网络规模可以提高方程求解精度,但也增加了网络训练迭代时间,在固定训练时间下并非神经网络规模越大效果越好.


关键词: Burgers-Fisher方程,  基于物理信息的神经网络,  规律信息,  数值信息,  数据驱动,  可解释性,  训练平衡度 
Fig.1 Schematic of PINN dealing with physical problems
Fig.2 Logical explanation of solving PDE with PINN
Fig.3 Schematic of PINN solving Burgers-Fisher equation
$ x $ $ t $ 参数情况1 参数情况2
$ {R_{{\text{Pre}}}} $/10?1 $ {R_{{\text{Exa}}}} $/10?1 $ {E_{{\text{Abs}}}} $/10?3 $ {R_{{\text{Pre}}}} $/10?1 $ {R_{{\text{Exa}}}} $/10?1 $ {E_{{\text{Abs}}}} $/10?4
0.0 0.0 4.9957 5.0000 0.4296 7.0735 7.0711 2.4521
0.1 1.0 5.2330 5.2436 1.0540 8.5860 8.5756 10.3780
0.2 2.0 5.4697 5.4860 1.6249 9.4012 9.4095 8.2731
0.3 3.0 5.7092 5.7261 1.6925 9.7709 9.7750 4.0752
0.4 4.0 5.9568 5.9628 0.6045 9.9170 9.9172 0.2986
0.5 5.0 6.2043 6.1952 0.9111 9.9715 9.9700 1.5283
0.6 6.0 6.4406 6.4222 1.8459 9.9921 9.9892 2.9022
0.7 7.0 6.6602 6.6430 1.7280 10.0000 9.9961 3.8970
0.8 8.0 6.8613 6.8568 0.4469 10.0030 9.9986 3.9810
0.9 9.0 7.0435 7.0630 1.9533 10.0020 9.9995 2.7907
1.0 10.0 7.2075 7.2611 5.3639 10.0000 9.9998 0.2772
Tab.1 Prediction results of diagonal data coordinates in grid sampling
Fig.4 Three-dimensional surface of equation solution on two cases
Fig.5 Change of absolute error with number of epochs of neural network training
$ {S_{{\text{Net}}}} $ $ {E_{{\text{Max}}}} $/10?2 $ {E_{{\text{Min}}}} $/10?6 $ {E_{{\text{Mea}}}} $/10?3 $ {E_{{\text{Sta}}}} $/10?3 $ {T_{{\text{Tim}}}} $/101
L2N10 1.7252 1.5875 5.0858 3.6818 0.8563
L2N20 1.1476 0.5662 2.6179 2.1679 1.0698
L2N40 0.9774 0.5504 1.8555 1.6741 1.2882
L4N10 0.7521 0.5027 2.2651 1.6853 1.2023
L4N20 0.6586 0.3417 1.5532 1.2726 1.5605
L4N40 0.4815 0.3338 1.1123 0.9160 2.2252
L6N10 0.4890 0.3775 1.3585 1.0056 1.6617
L6N20 0.3286 0.1570 0.7378 0.6244 2.1818
L6N40 0.3082 0.1371 0.6828 0.5584 3.0004
Tab.2 Descriptive statistic of absolute error of predicted values with scale of neural networks under a fixed epoch
$ {S_{{\text{Net}}}} $ $ {E_{{\text{Max}}}} $/10?2 $ {E_{{\text{Min}}}} $/10?6 $ {E_{{\text{Mea}}}} $/10?3 $ {E_{{\text{Sta}}}} $/10?3 $ {T_{{\text{Num}}}} $/102
L2N10 1.3816 1.5199 4.0196 2.9153 11.7200
L2N20 1.3958 1.1166 3.0385 2.5699 9.3000
L2N40 1.2500 0.9080 2.9150 2.3720 6.6600
L4N10 1.0755 0.8524 3.3827 2.4212 8.4700
L4N20 0.8338 0.8027 2.3934 1.7815 6.7100
L4N40 0.9156 0.6179 2.4211 1.8794 5.0600
L6N10 0.8874 1.3987 2.8323 2.0030 6.2000
L6N20 0.6949 0.5106 2.2007 1.5836 4.8400
L6N40 0.8039 0.7987 2.1866 1.6607 3.5000
Tab.3 Descriptive statistic of absolute error of predicted values with scale of neural networks under a fixed training time
$ {B_{{\text{Sam}}}} $ 方程内部 方程边缘 方程整体
$ {E_{{\text{Mea}}}} $/10?3 $ {E_{{\text{Sta}}}} $/10?3 $ {E_{{\text{Mea}}}} $/10?3 $ {E_{{\text{Sta}}}} $/10?3 $ {E_{{\text{Mea}}}} $/10?3 $ {E_{{\text{Sta}}}} $/10?3
3-0 907.00 82.20 838.00 118.00 872.00 107.00
0-1 2.63 2.30 1.75 1.54 2.19 2.04
3-1 1.59 1.22 1.56 1.36 1.57 1.30
6-0 907.00 82.90 839.00 118.00 873.00 108.00
0-2 2.50 2.07 1.65 1.47 2.08 1.88
6-2 1.55 1.19 1.56 1.35 1.55 1.28
9-0 904.00 82.20 836.00 118.00 870.00 107.00
0-3 2.35 1.98 1.50 1.19 1.92 1.72
9-3 1.41 1.09 1.47 1.25 1.44 1.18
Tab.4 Descriptive statistic of absolute error of predicted values with different training sampling balance
$ {B_{{\text{Int}}}} $ 方程内部 方程边缘 方程整体
$ {E_{{\text{Mea}}}} $/10?4 $ {E_{{\text{Sta}}}} $/10?4 $ {E_{{\text{Mea}}}} $/10?4 $ {E_{{\text{Sta}}}} $/10?4 $ {E_{{\text{Mea}}}} $/10?4 $ {E_{{\text{Sta}}}} $/10?4
1∶1 8.44 3.78 7.12 4.10 7.78 4.01
10∶1 7.73 3.43 8.11 4.43 7.92 5.14
1∶10 8.67 5.33 6.81 4.74 7.74 5.14
50∶1 7.27 2.42 6.41 3.20 6.84 2.93
1∶50 5.45 1.63 4.66 1.68 5.05 1.77
100∶1 13.00 4.06 11.10 5.45 1.20 4.95
1∶100 7.97 2.57 5.92 2.12 6.95 2.69
Tab.5 Descriptive statistic of absolute error of predicted values with different training intensity balance
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