METHODS AND APPLICATIONS OF PHYSICAL INFORMATION DEEP LEARNING IN WAVE NUMERICAL SIMULATION
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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.Abstract: In recent years, physics-informed deep learning methods based on prior data fusion to solve forward and inverse problems based on partial differential equations (PDEs) have become a cross-disciplinary hotspot. This paper clarifies the mathematical concept and implementation of physics-informed neural networks (PINN) for the earthquake engineering numerical simulation of waves. Taking the one-dimensional fluctuation of passive term as an example, the relevant theoretical model of PINN is constructed. The feasibility of physics-driven deep learning methods in solving fluctuation problems is verified by comparing with analytical solutions and finite difference methods. The relative
$ {\mathcal{L}}_{2} $ norm errors of the wave field simulated by PINN method and other numerical algorithms are analyzed. The physical driven deep learning method combined with sparse initial wave field data formed by spectral element method is used to numerically simulate two-dimensional fluctuation problem. Typical working conditions such as free boundary conditions and undulating ground surface are realized, and the distribution characteristics of time series wave field are given. Different initial conditions are changed to test the generalization accuracy of the neural network, and a transfer learning method was proposed to significantly improve the training efficiency of the network. By using transfer learning, wave fields at different source locations in infinite media can be directly predicted with high accuracy. Comparing with the results of spectral element method, the reliability of the proposed method is verified to simulate the wave propagation of homogeneous site, spatial inhomogeneity and complex terrain site fluctuation. The results show that the physical-driven deep learning method has the advantages of meshless and fine-grained numerical simulation, and can realize the numerical simulation conditions such as free surface and side/bottom boundary wave field transmission. -
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] 
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