留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

广义乘子法求解构造变分问题的神经网络方法

欧阳晔 江巍 吴怡 冯强 郑宏

欧阳晔, 江巍, 吴怡, 冯强, 郑宏. 广义乘子法求解构造变分问题的神经网络方法[J]. 工程力学, 2023, 40(11): 11-20. doi: 10.6052/j.issn.1000-4750.2022.05.0488
引用本文: 欧阳晔, 江巍, 吴怡, 冯强, 郑宏. 广义乘子法求解构造变分问题的神经网络方法[J]. 工程力学, 2023, 40(11): 11-20. doi: 10.6052/j.issn.1000-4750.2022.05.0488
OUYANG Ye, JIANG Wei, WU Yi, FENG Qiang, ZHENG Hong. NEURAL NETWORK METHOD FOR CONSTRUCTIVE VARIATIONAL PROBLEMS BY GENERALIZED MULTIPLIER METHOD[J]. Engineering Mechanics, 2023, 40(11): 11-20. doi: 10.6052/j.issn.1000-4750.2022.05.0488
Citation: OUYANG Ye, JIANG Wei, WU Yi, FENG Qiang, ZHENG Hong. NEURAL NETWORK METHOD FOR CONSTRUCTIVE VARIATIONAL PROBLEMS BY GENERALIZED MULTIPLIER METHOD[J]. Engineering Mechanics, 2023, 40(11): 11-20. doi: 10.6052/j.issn.1000-4750.2022.05.0488

广义乘子法求解构造变分问题的神经网络方法

doi: 10.6052/j.issn.1000-4750.2022.05.0488
基金项目: 国家自然科学基金项目(52079070);三峡库区地质灾害教育部重点实验室开放基金项目(2020KDZ10)
详细信息
    作者简介:

    欧阳晔(1996−),男,湖北武汉人,硕士生,主要从事土木工程物理数学模拟方法研究(E-mail: ouyangye123@qq.com)

    吴 怡(1999−),男,吉林长春人,硕士生,主要从事土木工程物理数学模拟方法研究(E-mail: 1743036363@qq.com)

    冯 强(1979−),男,湖北麻城人,副教授,博士,硕导,主要从事土木工程物理数学模拟方法研究(E-mail: qiangf2000@163.com)

    郑 宏(1964−),男,湖北南漳人,教授,博士,博导,主要从事土木工程物理数学模拟方法研究(E-mail: hzheng@whrsm.ac.cn)

    通讯作者:

    江 巍(1981−),男,湖北松滋人,教授,博士,博导,主要从事土木工程物理数学模拟方法研究(E-mail: jiangweilion@163.com)

  • 中图分类号: O34

NEURAL NETWORK METHOD FOR CONSTRUCTIVE VARIATIONAL PROBLEMS BY GENERALIZED MULTIPLIER METHOD

  • 摘要: 边界条件的施加是求解偏微分方程定解问题的重要步骤。神经网络方法求解偏微分方程定解问题时,将原问题转化为对应的构造变分问题后,损失函数是包含控制方程与边界条件的泛函。采用经典罚函数法及其改进方法施加边界条件时,罚因子的取值直接影响计算精度和求解效率;直接采用Lagrange乘子法施加边界条件,计算结果可能偏离原问题最优解。为破解此局限性,使用广义乘子法施加边界条件。基于神经网络获得原问题的预测解,再使用广义乘子法构建神经网络的损失函数并计算损失值,利用梯度下降法进行参数寻优,判断损失值是否满足要求;不满足则更新罚因子与乘子后再进行求解直至损失满足要求。数值算例的计算结果表明:与采用经典罚函数法、L1精确罚函数法和Lagrange乘子法施加边界条件构造的神经网络相比,该文提出的方法具有更好的数值精度和更高的求解效率,且求解过程更加稳定。
  • 图  1  两种常用的神经网络激活函数及其导数

    Figure  1.  Two widely used activation function in neural network and their derivatives

    图  2  配点示意图

    Figure  2.  Schematic diagram of collocation points

    图  3  物理信息网络示意图

    Figure  3.  Diagram of Physics Informed Neutral Network

    图  4  Kirchhoff矩形薄板示意图

    Figure  4.  Diagram of a Kirchhoff thin rectangular plate

    图  5  四种神经网络模型挠度结果的绝对误差平均值

    Figure  5.  Mean value of the absolute deflection errors resulted by four neural network models

    图  6  本文算法求解的薄板挠度云图

    Figure  6.  Deflect distribution of thin plate resulted by the proposed algorithm

    图  7  平面简支梁示意图

    Figure  7.  Diagram of a simple supported beam

    图  8  采用不同神经网络计算的平面弯曲梁挠度曲线

    Figure  8.  Deflection curve of plane bending beam by different neural network models

    图  9  采用不同神经网络计算的误差-耗时曲线

    Figure  9.  Error and time-consuming curves by different neural network models

    图  10  不同结构神经网络求解的误差-耗时曲线

    Figure  10.  Error and time-consuming curves by different neural network structures

    图  11  不同配点数与隐藏层神经元数量组合求解的误差

    Figure  11.  Error resulted by different combinations of collocation points and neurons in the hidden layer

    表  1  Kirchhoff薄板算例罚因子、乘子更新值

    Table  1.   The updating of penalty factor and multipliers in Kirchhoff thin plate example

    更新
    次数
    罚因子M乘子
    k1k2k3k4k5k6k7k8
    011.0001.0001.0001.0001.0001.0001.0001.000
    120.7500.7490.7520.7490.7490.7480.7470.749
    240.6790.6780.6830.6780.6780.6770.6760.678
    380.6490.6480.6550.6500.4470.6470.6460.648
    4160.6270.6400.6360.6380.6260.6410.6260.643
    5320.6220.6350.6320.6270.6210.6290.6200.640
    下载: 导出CSV

    表  2  简支梁算例罚因子、乘子更新值

    Table  2.   The updating of penalty factor and multipliers in simple supported beam example

    更新次数罚因子M乘子
    k1k2k3k4
    051.001.001.001.00
    150.360.640.971.35
    250.140.350.981.47
    3250.060.160.951.58
    41250.010.050.961.62
    下载: 导出CSV
  • [1] 姜礼尚, 陈亚浙, 刘西垣, 等. 数学物理方程讲义[M]. 3版. 北京: 高等教育出版社, 2007: 1 − 2.

    JIANG Lishang, CHEN Yazhe, LIU Xiyuan, et al. Equations of mathematical physics [M]. 3rd ed. Beijing: Higher Education Press, 2007: 1 − 2. (in Chinese)
    [2] 王昀卓. 求解偏微分方程的神经网络方法[D]. 合肥: 中国科学技术大学, 2021.

    WANG Yunzhuo. Neural network for solving partial differential equations [D]. Hefei: University of Science and Technology of China, 2021. (in Chinese)
    [3] 焦李成, 杨淑媛, 刘芳, 等. 神经网络七十年: 回顾与展望[J]. 计算机学报, 2016, 39(8): 1697 − 1717.

    JIAO Licheng, YANG Shuyuan, LIU Fang, et al. Seventy years beyond neural networks: Retrospect and prospect [J]. Chinese Journal of Computers, 2016, 39(8): 1697 − 1717. (in Chinese)
    [4] XIE Y L, HE M J, MA T S, et al. Optimal distributed parallel algorithms for deep learning framework TensorFlow [J]. Applied Intelligence, 2022, 52(4): 3880 − 3900. doi: 10.1007/s10489-021-02588-9
    [5] DAI H L, PENG X, SHI X H, et al. Reveal training performance mystery between TensorFlow and Pytorch in the single GPU environment [J]. Science China Information Sciences, 2022, 65: 112103. doi: 10.1007/s11432-020-3182-1
    [6] 叶继红, 杨振宇. 基于生成式对抗网络的风场生成研究[J]. 工程力学, 2021, 38(10): 1 − 11. doi: 10.6052/j.issn.1000-4750.2020.10.0721

    YE Jihong, YANG Zhenyu. Research on generation of wind fields based on GAN [J]. Engineering Mechanics, 2021, 38(10): 1 − 11. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.10.0721
    [7] 许泽坤, 陈隽. 非线性结构地震响应的神经网络算法[J]. 工程力学, 2021, 38(9): 133 − 145. doi: 10.6052/j.issn.1000-4750.2020.09.0645

    XU Zekun, CHEN Jun. Neural network algorithm for nonlinear structural seismic response [J]. Engineering Mechanics, 2021, 38(9): 133 − 145. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.09.0645
    [8] 程诗焱, 韩建平, 于晓辉, 等. 基于BP神经网络的RC框架结构地震易损性曲面分析: 考虑地震动强度和持时的影响[J]. 工程力学, 2021, 38(12): 107 − 117. doi: 10.6052/j.issn.1000-4750.2020.11.0837

    CHENG Shiyan, HAN Jianping, YU Xiaohui, et al. Seismic fragility surface analysis of RC frame structures based on BP neural networks: Accounting for the effects of ground motion intensity and duration [J]. Engineering Mechanics, 2021, 38(12): 107 − 117. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.11.0837
    [9] 赵林鑫, 江守燕, 杜成斌. 基于SBFEM和机器学习的薄板结构缺陷反演[J]. 工程力学, 2021, 38(6): 36 − 46. doi: 10.6052/j.issn.1000-4750.2020.06.0416

    ZHAO Linxin, JIANG Shouyan, DU Chengbin. Flaws detection in thin plate structures based on SBFEM and machine learning [J]. Engineering Mechanics, 2021, 38(6): 36 − 46. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.06.0416
    [10] 郑秋怡, 周广东, 刘定坤. 基于长短时记忆神经网络的大跨拱桥温度-位移相关模型建立方法[J]. 工程力学, 2021, 38(4): 68 − 79. doi: 10.6052/j.issn.1000-4750.2020.05.0323

    ZHENG Qiuyi, ZHOU Guangdong, LIU Dingkun. Method of modeling temperature-displacement correlation for long-span arch bridges based on long short-term memory neural networks [J]. Engineering Mechanics, 2021, 38(4): 68 − 79. (in Chinese) doi: 10.6052/j.issn.1000-4750.2020.05.0323
    [11] E W N, YU B. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems [J]. Communications in Mathematics and Statistics, 2018, 6(1): 1 − 12. doi: 10.1007/s40304-018-0127-z
    [12] SIRIGNANO J, SPILIOPOULOS K. DGM: A deep learning algorithm for solving partial differential equations [J]. Journal of Computational Physics, 2018, 375: 1339 − 1364. doi: 10.1016/j.jcp.2018.08.029
    [13] SAMANIEGO E, ANITESCU C, GOSWAMI S, et al. An energy approach to the solution of partial differential equations in computational mechanics via machine learning: Concepts, implementation and applications [J]. Computer Methods in Applied Mechanics and Engineering, 2020, 362: 112790. doi: 10.1016/j.cma.2019.112790
    [14] RAISSI M, PERDIKARIS P, KARNIADAKIS G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations [J]. Journal of Computational Physics, 2019, 378: 686 − 707. doi: 10.1016/j.jcp.2018.10.045
    [15] RAISSI M, YAZDANI A, KARNIADAKIS G E. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations [J]. Science, 2020, 367(6481): 1026 − 1030. doi: 10.1126/science.aaw4741
    [16] LU L, MENG X H, MAO Z P, et al. DeepXDE: A deep learning library for solving differential equations [J]. SIAM Review, 2021, 63(1): 208 − 228. doi: 10.1137/19M1274067
    [17] 郭宏伟, 庄晓莹. 采用两步优化器的深度配点法与深度能量法求解薄板弯曲问题[J]. 固体力学学报, 2021, 42(3): 249 − 266. doi: 10.19636/j.cnki.cjsm42-1250/o3.2021.029

    GUO Hongwei, ZHUANG Xiaoying. The application of deep collocation method and deep energy method with a two-step optimizer in the bending analysis of Kirchhoff thin plate [J]. Chinese Journal of Solid Mechanics, 2021, 42(3): 249 − 266. (in Chinese) doi: 10.19636/j.cnki.cjsm42-1250/o3.2021.029
    [18] 黄钟民, 谢臻, 张易申, 等. 面内变刚度薄板弯曲问题的挠度-弯矩耦合神经网络方法[J]. 力学学报, 2021, 53(9): 2541 − 2553. doi: 10.6052/0459-1879-21-273

    HUANG Zhongmin, XIE Zhen, ZHANG Yishen, et al. Deflection-bending moment coupling neural network method for the bending problem of thin plates with in-plane stiffness gradient [J]. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(9): 2541 − 2553. (in Chinese) doi: 10.6052/0459-1879-21-273
    [19] 唐明健, 唐和生. 基于物理信息的深度学习求解矩形薄板力学正反问题[J]. 计算力学学报, 2022, 39(1): 120 − 128. doi: 10.7511/jslx20201110003

    TANG Mingjian, TANG Hesheng. A physics-informed deep learning method for solving forward and inverse mechanics problems of thin rectangular plates [J]. Chinese Journal of Computational Mechanics, 2022, 39(1): 120 − 128. (in Chinese) doi: 10.7511/jslx20201110003
    [20] HE J C, LI L, XU J C, et al. ReLU deep neural networks and linear finite elements [J]. Journal of Computational Mathematics, 2020, 38(3): 502 − 527. doi: 10.4208/jcm.1901-m2018-0160
    [21] 黄钟民, 陈思亚, 陈卫, 等. 薄板弯曲问题的神经网络方法[J]. 固体力学学报, 2021, 42(6): 697 − 706.

    HUANG Zhongmin, CHEN Siya, CHEN Wei, et al. Neural network method for thin plate bending problem [J]. Chinese Journal of Solid Mechanics, 2021, 42(6): 697 − 706. (in Chinese)
    [22] 张光澄. 非线性最优化计算方法[M]. 北京: 高等教育出版社, 2005: 292 − 309.

    ZHANG Guangcheng. Computational methods for nonlinear optimization [M]. Beijing: Higher Education Press, 2005: 292 − 309. (in Chinese)
    [23] 徐慧福. 混合约束不可微非线性规划的L1-精确罚函数法[J]. 宁波大学学报, 1994, 7(2): 1 − 9.

    XU Huifu. L1 exact penalty methods for inequality and equality constrained programming [J]. Journal of Ningbo University, 1994, 7(2): 1 − 9. (in Chinese)
    [24] 李海滨, 段志信. 约束非线性规划问题的L1精确罚函数神经网络方法[J]. 电子学报, 2009, 37(1): 229 − 234. doi: 10.3321/j.issn:0372-2112.2009.01.043

    LI Haibin, DUAN Zhixin. An L1 exact penalty function neural network method for constraint nonlinear programming problems [J]. Acta Electronica Sinica, 2009, 37(1): 229 − 234. (in Chinese) doi: 10.3321/j.issn:0372-2112.2009.01.043
    [25] 陈珊珊, 楼旭阳, 崔宝同. 参数非线性规划问题的L1精确罚函数神经网络方法分析[J]. 计算机应用与软件, 2014, 31(7): 277 − 279, 315. doi: 10.3969/j.issn.1000-386x.2014.07.071

    CHEN Shanshan, LOU Xuyang, CUI Baotong. Analysing L1 exact penalty function neural networks method of parametric nonlinear programming problems [J]. Computer Applications and Software, 2014, 31(7): 277 − 279, 315. (in Chinese) doi: 10.3969/j.issn.1000-386x.2014.07.071
    [26] 王尚长, 杨格, 吴斌, 等. 基于内点法和拉格朗日乘子法的混合试验冗余作动器控制方法[J]. 振动与冲击, 2021, 40(12): 23 − 30, 37.

    WANG Shangzhang, YANG Ge, WU Bin, et al. A redundant actuator control method for hybrid simulation based on the interior point and the Lagrange multiplier [J]. Journal of Vibration and Shock, 2021, 40(12): 23 − 30, 37. (in Chinese)
    [27] 蒋昂波, 王维维. ReLU激活函数优化研究[J]. 传感器与微系统, 2018, 37(2): 50 − 52.

    JIANG Angbo, WANG Weiwei. Research on optimization of ReLU activation function [J]. Transducer and Microsystem Technologies, 2018, 37(2): 50 − 52. (in Chinese)
    [28] 徐芝纶. 弹性力学简明教程[M]. 5版. 北京: 高等教育出版社, 2018: 214 − 216.

    XU Zhilun. Concise course in elasticity [M]. 5th ed. Beijing: Higher Education Press, 2018: 214 − 216. (in Chinese)
  • 加载中
图(11) / 表(2)
计量
  • 文章访问数:  127
  • HTML全文浏览量:  43
  • PDF下载量:  45
  • 被引次数: 0
出版历程
  • 收稿日期:  2022-05-29
  • 修回日期:  2022-10-07
  • 网络出版日期:  2022-10-29
  • 刊出日期:  2023-11-06

目录

    /

    返回文章
    返回