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基于神经网络的偏微分方程求解方法研究综述

查文舒 李道伦 沈路航 张雯 刘旭亮

查文舒, 李道伦, 沈路航, 张雯, 刘旭亮. 基于神经网络的偏微分方程求解方法研究综述. 力学学报, 待出版 doi: 10.6052/0459-1879-21-617
引用本文: 查文舒, 李道伦, 沈路航, 张雯, 刘旭亮. 基于神经网络的偏微分方程求解方法研究综述. 力学学报, 待出版 doi: 10.6052/0459-1879-21-617
Zha Wenshu, Li Daolun, Shen Luhang, Zhang Wen, Liu Xuliang. Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-21-617
Citation: Zha Wenshu, Li Daolun, Shen Luhang, Zhang Wen, Liu Xuliang. Review of neural network-based methods for solving partial differential equations. Chinese Journal of Theoretical and Applied Mechanics, in press doi: 10.6052/0459-1879-21-617

基于神经网络的偏微分方程求解方法研究综述

doi: 10.6052/0459-1879-21-617
基金项目: 国家自然科学基金项目(1217020361)资助.
详细信息
    作者简介:

    查文舒, 副研究员, 主要研究方向: 流动机理及数值模拟研究. E-mail:wszha@hfut.edu.cn

    李道伦, 教授, 主要研究方向: 流动机理及数值模拟研究. E-mail:ldaol@hfut.edu.cn

  • 中图分类号: O241

REVIEW OF NEURAL NETWORK-BASED METHODS FOR SOLVING PARTIAL DIFFERENTIAL EQUATIONS

  • 摘要: 神经网络作为一种强大的信息处理工具在计算机视觉, 生物医学, 油气工程领域得到广泛应用, 引发多领域技术变革. 深度学习网络具有非常强的学习能力, 不仅能发现物理规律, 还能求解偏微分方程(PDE). 近年来基于深度学习的PDE求解已是研究新热点. 遵循于传统PDE解析解、PDE数值解术语, 本文称用神经网络进行PDE求解的方法为PDE智能求解方法或PDE神经网络求解方法. 本文首先简要介绍PDE智能求解发展历程, 然后从反演未知PDE与求解已知PDE两个角度展开讨论, 重点讨论已知PDE的求解方法. 根据神经网络中损失函数的构建方式, 将PDE求解方法分为三大类: 第一类是数据驱动, 主要从数据中学习PDE, 可以应用于恢复方程、参数反演等; 第二类是物理约束, 即在数据驱动的基础上, 辅以物理约束, 在损失函数中加入控制方程等物理规律, 减少网络对标签数据的依赖, 大幅提高泛化能力与应用价值; 第三类物理驱动(纯物理约束), 完全不使用标签数据, 仅通过物理规律求解PDE, 目前仅适用于简单偏微分方程. 本文从这三个方面介绍PDE智能求解的研究进展, 涉及全连接神经网络、卷积神经网络、循环神经网络等多种网络结构. 最后总结PDE智能求解的研究进展, 给出相应的应用场景以及未来研究展望.

     

  • 图  1  基于PDE智能求解方法的两种技术路线

    Figure  1.  Two technical routes based on PDE intelligent solution method

    图  2  3 D-PDE-Net网络结构示意图[74]

    Figure  2.  The schematic diagram of a δt-block[74]

    图  3  有H个隐藏层、N个时间间隔的半线性抛物型偏微分方程的网络结构(修改自文献[29])

    Figure  3.  Illustration of the network architecture for solving semilinear parabolic PDEs with H hidden layers for each sub-network and N time intervals. (Modified from literature[29])

    图  4  Sigmoid, tanh, ReLU和Leaky-ReLU的对应变量$a$的激活函数

    Figure  4.  Sigmoid, tanh, ReLU and Leaky-ReLU activation functions for various values of $a$

    图  5  TgNN模型的网络结构[87]

    Figure  5.  Structure of the TgNN model[87]

    图  6  智能求解得到的压力分布和井底压力图[89]

    Figure  6.  Pressure distribution and BHP obtained by intelligent solution [89]

    图  7  基于“硬边界约束”的FC-NN框架(修改自文献[94])

    Figure  7.  Schematic diagram of FC-NN framework based on “hard boundary constraint” (Modified from literature[94])

    图  8  贝叶斯损失函数约束下的DenseED-c16网络(修改自文献[96])

    Figure  8.  DenseED-c16 network with Bayesian loss function constraints (Modified from literature[96])

    图  9  物理约束下的稠密卷积编解码器网络(修改自文献[98])

    Figure  9.  Dense convolutional encoder-decoder network as the deterministic surrogate (Modified from literature[98])

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  • 录用日期:  2022-01-10
  • 网络出版日期:  2022-01-10

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