EI、Scopus 收录
中文核心期刊

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

物理驱动深度学习波动数值模拟方法及应用

陈苏 丁毅 孙浩 赵密 王进廷 李小军

陈苏, 丁毅, 孙浩, 赵密, 王进廷, 李小军. 物理驱动深度学习波动数值模拟方法及应用. 力学学报, 2023, 55(1): 272-282 doi: 10.6052/0459-1879-22-401
引用本文: 陈苏, 丁毅, 孙浩, 赵密, 王进廷, 李小军. 物理驱动深度学习波动数值模拟方法及应用. 力学学报, 2023, 55(1): 272-282 doi: 10.6052/0459-1879-22-401
Chen Su, Ding Yi, Sun Hao, Zhao Mi, Wang Jinting, Li Xiaojun. Methods and applications of physical information deep learning in wave numerical simulation. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 272-282 doi: 10.6052/0459-1879-22-401
Citation: Chen Su, Ding Yi, Sun Hao, Zhao Mi, Wang Jinting, Li Xiaojun. Methods and applications of physical information deep learning in wave numerical simulation. Chinese Journal of Theoretical and Applied Mechanics, 2023, 55(1): 272-282 doi: 10.6052/0459-1879-22-401

物理驱动深度学习波动数值模拟方法及应用

doi: 10.6052/0459-1879-22-401
基金项目: 国家自然科学基金(52192675)和国家自然科学地震联合基金(U1839202)资助项目
详细信息
    作者简介:

    通讯作者: 陈苏, 教授, 主要研究方向为地震工程与人工智能交叉. E-mail: chensuchina@126.com

    通讯作者:

    李小军, 教授, 主要研究方向为地震工程. E-mail: beerli@vip.sina.com

  • 中图分类号: P315

METHODS AND APPLICATIONS OF PHYSICAL INFORMATION DEEP LEARNING IN WAVE NUMERICAL SIMULATION

  • 摘要: 近年来, 物理先验融合数据的深度学习方法求解以偏微分方程为理论基础的正反演问题已成为交叉学科热点. 针对地震工程波动数值模拟, 本文阐明了物理驱动深度学习方法PINN的数学概念及实现方式, 以无源项一维波动为例, 开展了相关理论模型构建, 并与解析解及有限差分方法进行对比, 分析了PINN方法与其他数值算法模拟波场的相对范数误差, 验证了物理驱动深度学习方法求解波动问题的可行性. 采用物理驱动深度学习方法并结合谱元法形成的稀疏初始波场数据, 开展了二维波动数值模拟, 实现了自由边界条件及起伏地表等典型工况的模拟, 并给出了时序波场分布特性. 更换不同的初始条件, 测试了神经网络的泛化精度, 提出可显著提高网络训练效率的迁移学习方法. 通过与谱元法的结果对比, 验证了本文方法模拟均质场地、空间不均匀及复杂地形场地波动问题的可靠性. 结果表明, 物理驱动深度学习方法具备无网格、精细化模拟等优势, 并可实现自由地表及侧边界波场透射等数值模拟条件.

     

  • 图  1  无源项一维波动问题: 不同损失项采样点时空分布

    Figure  1.  One-dimensional fluctuation problem of passive term: spatiotemporal distribution of sampling points for different loss terms

    图  2  无源项一维波动问题: 神经网络训练过程中不同损失项的演化

    Figure  2.  One-dimensional fluctuation problem of passive terms: evolution of different loss terms during neural network training

    图  3  无源项一维波动问题: 解析解与PINN预测的结果对比

    Figure  3.  One-dimensional fluctuation problem with passive term: comparison between analytical solution and PINN prediction

    图  4  求解二维波动方程的物理信息神经网络架构图

    Figure  4.  Physical information neural network architecture diagram for solving the 2 D wave equation

    图  5  无限均匀介质内源波动计算模型

    Figure  5.  Calculation model of endogenous fluctuation in infinite uniform medium

    图  6  无限均匀介质波动问题不同损失项时空采样点分布

    Figure  6.  Spatio-temporal sampling points distribution of different loss terms for infinite uniform medium fluctuation problem

    图  7  PINN方法与谱元方法模拟波场之间的相对$ {\mathcal{L}}_{2} $范数误差与决定系数$ {R}^{2} $

    Figure  7.  Relative $ {\mathcal{L}}_{2} $ norm error and determination coefficient $ {R}^{2} $ between the results of PINN method and spectral element method

    图  8  无限均匀介质波动问题: 谱元法与PINN模拟的波场快照对比

    Figure  8.  Wave field snapshot of spectral element method and PINN simulation

    图  9  PINN方法使用迁移学习与谱元法模拟波场之间的相对$ {\mathcal{L}}_{2} $范数误差与决定系数$ {R}^{2} $

    Figure  9.  Relative $ {\mathcal{L}}_{2} $ norm error and determination coefficient $ {R}^{2} $ between the results of PINN method using transfer learning and spectral element method

    图  10  谱元法与PINN方法使用迁移学习得到的波场快照对比

    Figure  10.  Wave field snapshots obtained using spectral element method and PINN method using transfer learning

    图  11  空间不均匀介质物理波速分布

    Figure  11.  Physical wave velocity distribution in spatially inhomogeneous media

    图  12  PINN方法与谱元方法模拟波场之间的相对$ {\mathcal{L}}_{2} $范数误差与决定系数$ {R}^{2} $

    Figure  12.  Relative $ {\mathcal{L}}_{2} $ norm error and determination coefficient $ {R}^{2} $ between the results of PINN method and spectral element method

    图  13  空间不均匀介质波动问题: 谱元法与PINN模拟的波场快照

    Figure  13.  Spatially inhomogeneous media fluctuation problem: wave field snapshots simulated by spectral element method and PINN

    图  14  起伏地形内源波动问题计算模型

    Figure  14.  Computational model for endogenous fluctuation problems in undulating terrain

    图  15  起伏地形内源波动问题不同损失项时空采样点分布

    Figure  15.  Distribution of spatial and temporal sampling points for different loss terms for endogenous fluctuation problems

    图  16  PINN方法与谱元方法模拟波场之间的相对$ {\mathcal{L}}_{2} $范数误差与决定系数$ {R}^{2} $

    Figure  16.  Relative $ {\mathcal{L}}_{2} $ norm error and determination coefficient $ {R}^{2} $ between the results of PINN method and spectral element method

    图  17  起伏地表内源波动问题: 谱元法与PINN模拟的波场快照

    Figure  17.  Internal fluctuation problem of undulating surface: wave field snapshots of spectral element method and PINN simulation

    表  1  PINN方法在不同领域中的应用

    Table  1.   Application of PINN method in different fields

    领域应用场景时间文献
    流体动力学无需标签数据对流体流动过程进行建模2020[15]
    多孔介质中的地下水流动问题2020[16]
    提出一种混合变量格式PINN模拟低雷诺数下的稳态和瞬态层流2020[17]
    不可压缩Navier-Stokes流2020[18]
    基于Navier-Stokes方程的流场
    重建
    2021[19]
    热传导具有内热源的二维稳态导热方程和平板间二维稳态对流传热方程2021[20]
    热传导方程的正问题及逆问题2021[21]
    地球物理通过求解Eikonal方程来预测波在非均匀介质中的传播时间2020[22]
    求解复杂二维声学介质中的压力响应2020[23]
    声波传播和全波形反演(FWIs)2022[24]
    利用亥姆霍兹方程定义损失函数实现波场重建反演(WRI)2022[25]
    固体力学识别和表征金属板表面断裂裂纹的问题2020[26]
    求解矩形薄板力学正反问题2022[27]
    无需标签数据求解二维平面应力问题与弹性波传播问题2021[28]
    将动量平衡与本构关系融入到PINN中, 并探讨了在线性弹性力学和非线性问题中的应用2021[29]
    生物医学心房颤动诊断问题2020[30]
    下载: 导出CSV
  • [1] 胡自多, 刘威, 雍学善等. 三维波动方程时空域混合网格有限差分数值模拟方法. 地球物理学报, 2021, 64(8): 2809-2828 (Hu Ziduo, Liu Wei, Yong Xueshan, et al. Mixed-grid finite-difference method for numerical simulation of 3D wave equation in the time-space domain. Chinese Journal of Geophysics, 2021, 64(8): 2809-2828 (in Chinese) doi: 10.6038/cjg2021O0296
    [2] 吴国忱, 李青阳, 吴建鲁等. 流固边界耦合介质高阶有限差分地震正演模拟方法. 地震学报, 2018, 40(1): 32-44 (Wu Guochen, Li Qingyang, Wu Jianlu, et al. High-order finite-difference seismic forward modeling method for fluid-solid boundary coupling media. Acta Seismologica Sinica, 2018, 40(1): 32-44 (in Chinese) doi: 10.11939/jass.20170010
    [3] 薛东川, 王尚旭, 焦淑静. 起伏地表复杂介质波动方程有限元数值模拟方法. 地球物理学进展, 2007(2): 522-529 (Xue Dongchuan, Wang Shangxu, Jiao Shujing. Wave equation finite-element modeling including rugged topography and complicated medium. Progress in Geophysics, 2007(2): 522-529 (in Chinese) doi: 10.3969/j.issn.1004-2903.2007.02.026
    [4] Liu YS, Teng JW, Lan HQ, et al. A comparative study of finite element and spectral element methods in seismic wavefield modeling. Geophysics, 2014, 79(2): T91-T104 doi: 10.1190/geo2013-0018.1
    [5] 李孝波, 薄景山, 齐文浩等. 地震动模拟中的谱元法. 地球物理学进展, 2014, 29(5): 2029-2039 (Li Xiaobo, Bo Jingshan, Qi Wenhao, et al. Spectral element method in seismic ground motion simulation. Progress in Geophysics, 2014, 29(5): 2029-2039 (in Chinese) doi: 10.6038/pg20140506
    [6] 刘少林, 杨顶辉, 徐锡伟等. 模拟地震波传播的三维逐元并行谱元法. 地球物理学报, 2021, 64(3): 993-1005 (Liu Shaolin, Yang Dinghui, Xu Xiwei, et al. Three-dimensional element-by-element parallel spectral-element method for seismic wave modeling. Chinese Journal of Geophysics, 2021, 64(3): 993-1005 (in Chinese)
    [7] 周红, 高孟潭, 俞言祥. SH波地形效应特征的研究. 地球物理学进展, 2010, 25(3): 775-782 (Zhou Hong, Gao Mengtan, Yu Yanxiang. A study of topographical effect on SH waves. Progress in Geophysics, 2010, 25(3): 775-782 (in Chinese) doi: 10.3969/j.issn.1004-2903.2010.03.005
    [8] 邢浩洁, 李鸿晶, 李小军. 一维波动有限元模拟中透射边界的时域稳定条件. 应用基础与工程科学学报, 2021, 29(3): 617-632 (Xing Haojie, Li Hongjing, Li Xiaojun. Time-domain stability conditions of multi-transmitting formula in one-dimensional finite-element simulation of wave motion. Journal of Basic Science and Engineering, 2021, 29(3): 617-632 (in Chinese) doi: 10.16058/j.issn.1005-0930.2021.03.008
    [9] 邢浩洁, 李小军, 刘爱文等. 波动数值模拟中的外推型人工边界条件. 力学学报, 2021, 53(5): 1480-1495 (Xing Haojie, Li Xiaojun, Liu Aiwen, et al. Extrapolation-type artificial boundary conditions in the numerical simulation of wave motion. Chinese Journal of Theoretical and Applied Mechanics, 2021, 53(5): 1480-1495 (in Chinese)
    [10] 章旭斌, 廖振鹏, 谢志南. 透射边界高频耦合失稳机理及稳定实现——SH波动. 地球物理学报, 2015, 58(10): 3639-3648 (Zhang Xubin, Liao Zhenpeng, Xie Zhinan. Mechanism of high frequency coupling instability and stable implementation for transmitting boundary—SH wave motion. Chinese Journal of Geophysics, 2015, 58(10): 3639-3648 (in Chinese) doi: 10.6038/cjg20151017
    [11] E WN, Yu B. The deep Ritz method: A deep learning-based numerical algorithm for solving variational problems. 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. Journal of Computational Physics, 2018, 375: 1339-1364 doi: 10.1016/j.jcp.2018.08.029
    [13] Raissi M, Perdikaris P, Karniadakis GE. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 2019, 378: 686-707 doi: 10.1016/j.jcp.2018.10.045
    [14] Lagaris IE, Likas A, Fotiadis DI. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 1998, 9(5): 987-1000 doi: 10.1109/72.712178
    [15] Sun LN, Gao H, Pan SW, et al. Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, 2020, 361: 112732 doi: 10.1016/j.cma.2019.112732
    [16] Guo HW, Zhuang XY, Liang D, et al. Stochastic groundwater flow analysis in heterogeneous aquifer with modified neural architecture search (NAS) based physics-informed neural networks using transfer learning. arXiv preprint, arXiv: 2010.12344, 2020
    [17] Rao CP, Sun H, Liu Y. Physics-informed deep learning for incompressible laminar flows. Theoretical and Applied Mechanics Letters, 2020, 10(3): 207-212 doi: 10.1016/j.taml.2020.01.039
    [18] Jin XW, Cai SZ, Li H, et al. NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics, 2020, 426: 109951
    [19] 尧少波, 何伟峰, 陈丽华等. 融合物理的神经网络方法在流场重建中的应用. 空气动力学学报, 2022, 40(5): 30-38

    Yao Shaobo, He Weifeng, Chen Lihua. Physics informed neural network in flowfield reconstruction. Acta Aerodynamica Sinica, 2022, 40(5): 30-38 (in Chinese))
    [20] 陆至彬, 瞿景辉, 刘桦等. 基于物理信息神经网络的传热过程物理场代理模型的构建. 化工学报, 2021, 72(3): 1496-1503 (Lu Zhibin, Qu Jinghui, Liu Hua, et al. Surrogate modeling for physical fields of heat transfer processes based on physics-informed neural network. CIESC Journal, 2021, 72(3): 1496-1503 (in Chinese) doi: 10.11949/0438-1157.20201879
    [21] 赵暾, 周宇, 程艳青等. 基于内嵌物理机理神经网络的热传导方程的正问题及逆问题求解. 空气动力学学报, 2021, 39(5): 19-26 (Zhao Tun, Zhou Yu, Cheng Yanqing, et al. Solving forward and inverse problems of the heat conduction equation using physics-informed neural networks. Acta Aerodynamica Sinica, 2021, 39(5): 19-26 (in Chinese) doi: 10.7638/kqdlxxb-2020.0176
    [22] Smith JD, Azizzadenesheli K, Ross ZE. Eikonet: Solving the eikonal equation with deep neural networks. IEEE Transactions on Geoscience and Remote Sensing, 2020, 59(12): 10685-10696
    [23] Moseley B, Markham A, Nissen-Meyer T. Solving the wave equation with physics-informed deep learning. arXiv preprint, arXiv: 2006.11894, 2020
    [24] Rasht-Behesht M, Huber C, Shukla K, et al. Physics-informed neural networks (PINNs) for wave propagation and full waveform inversions. Journal of Geophysical Research:Solid Earth, 2022, 127(5): e2021JB023120
    [25] Song C, Alkhalifah TA. wavefield reconstruction inversion via physics-informed neural networks. IEEE Transactions on Geoscience and Remote Sensing, 2022, 60: 1-12
    [26] Shukla K, Di Leoni PC, Blackshire J, et al. Physics-informed neural network for ultrasound nondestructive quantification of surface breaking cracks. Journal of Nondestructive Evaluation, 2020, 39(3): 1-20
    [27] 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. 计算力学学报, 2022, 39(1): 120-128 (Tang Mingjian, Tang Hesheng. A physics-informed deep learning method for solving forward and inverse mechanics problems of thin rectangular plates. Chinese Journal of Computational Mechanics, 2022, 39(1): 120-128 (in Chinese)
    [28] Rao CP, Sun H, Liu Y. Physics-informed deep learning for computational elastodynamics without labeled data. Journal of Engineering Mechanics, 2021, 147(8): 04021043 doi: 10.1061/(ASCE)EM.1943-7889.0001947
    [29] Haghighat E, Raissi M, Moure A, et al. A physics-informed deep learning framework for inversion and surrogate modeling in solid mechanics. Computer Methods in Applied Mechanics and Engineering, 2021, 379: 113741 doi: 10.1016/j.cma.2021.113741
    [30] Kissas G, Yang YB, Hwuang E, et al. Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks. Computer Methods in Applied Mechanics and Engineering, 2020, 358: 112623 doi: 10.1016/j.cma.2019.112623
    [31] Cuomo S, Di Cola VS, Giampaolo F, et al. Scientific machine learning through physics–informed neural networks: where we are and what’s next. Journal of Scientific Computing, 2022, 92(3): 1-62
    [32] 李野, 陈松灿. 基于物理信息的神经网络: 最新进展与展望. 计算机科学, 2022, 49(4): 254-262 (Li Ye, Cheng Songcan. Physics-informed neural networks: recent advances and prospects. Computer Science, 2022, 49(4): 254-262 (in Chinese)
    [33] Karniadakis GE, Kevrekidis IG, Lu L, et al. Physics-informed machine learning. Nature Reviews Physics, 2021, 3(6): 422-440 doi: 10.1038/s42254-021-00314-5
    [34] Cai SZ, Mao ZP, Wang ZC, et al. Physics-informed neural networks (PINNs) for fluid mechanics: a review. Acta Mechanica Sinica, 2021, 37(12): 1727-1738 doi: 10.1007/s10409-021-01148-1
    [35] Chen TP, Chen H. Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Transactions on Neural Networks, 1995, 6(4): 911-917 doi: 10.1109/72.392253
    [36] Sobol’ IM. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Computational Mathematics and Mathematical Physics, 1967, 7(4): 86-112 doi: 10.1016/0041-5553(67)90144-9
    [37] Komatitsch D, Tromp J. A perfectly matched layer absorbing boundary condition for the second-order seismic wave equation. Geophysical Journal International, 2003, 154(1): 146-153 doi: 10.1046/j.1365-246X.2003.01950.x
    [38] 王周, 李朝晖, 龙桂华等. 求解弹性波有限差分法中自由边界处理方法的对比. 工程力学, 2012, 29(4): 77-83 (Wang Zhou, Li Zhaohui, Long Guihua, et al. Comparison among implementations of free-surface boundary in elastic wave simulation using the finite-difference method. Engineering Mechanics, 2012, 29(4): 77-83 (in Chinese)
  • 加载中
图(17) / 表(1)
计量
  • 文章访问数:  625
  • HTML全文浏览量:  186
  • PDF下载量:  220
  • 被引次数: 0
出版历程
  • 收稿日期:  2022-08-30
  • 录用日期:  2022-11-25
  • 网络出版日期:  2022-12-01
  • 刊出日期:  2023-01-04

目录

    /

    返回文章
    返回