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 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, 2022, 54(3): 543-556.
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, 2022, 54(3): 543-556.
 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, 2022, 54(3): 543-556.

## 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. 近年来基于深度学习的偏微分方程求解已是研究新热点. 遵循于传统偏微分方程解析解、偏微分方程数值解术语, 本文称用神经网络进行偏微分方程求解的方法为偏微分方程智能求解方法或偏微分方程神经网络求解方法. 本文首先简要介绍偏微分方程智能求解发展历程, 然后从反演未知偏微分方程与求解已知偏微分方程两个角度展开讨论, 重点讨论已知偏微分方程的求解方法. 根据神经网络中损失函数的构建方式, 将偏微分方程求解方法分为3大类: 第1类是数据驱动, 主要从数据中学习偏微分方程, 可以应用于恢复方程、参数反演等; 第2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3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偏微分方程, 目前仅适用于简单偏微分方程. 本文从这3个方面介绍偏微分方程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. 最后总结偏微分方程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.

Abstract: Neural networks are widely used as a powerful information processing tool in the fields of computer vision, biomedicine, and oil-gas engineering, triggering technological changes. Due to the powerful learning ability, deep learning networks can not only discover physical laws but also solve partial differential equations (PDEs). In recent years, PDE solving based on deep learning has been a new research hotspot. Following the terms of traditional PDE analytical solution, this paper calls the method of solving PDE by neural network as PDE intelligent solution or PDE neural-network solution. This paper briefly introduces the development history of PDE intelligent solution, and then discusses the development of recovering unknown PDEs and solving known PDEs. The main focus of this paper is on a neural network solution method for a known PDE. It is divided into three categories according to the way of constructing loss functions. The first is data-driven method, which mainly learns PDEs from partially known data and can be applied to recovering physical equations, discovering unknown equations, parameter inversion, etc. The second is physical-constraint method, i.e., data-driven supplemented by physical constraints, which is manifested by adding physical laws such as governing equation to the loss function, thus reducing the network's reliance on labeled data and improving the generalization ability and application value. The third is physics-driven method (purely physical constraints), which solves PDEs by physical laws without any labeled data. However, such methods are currently only applied to solve simple PDEs and still need to be improved for complex physics. This paper introduces the research progress of intelligent solution of PDEs from these three aspects, involving various network structures such as fully-connected neural networks, convolutional neural networks, recurrent neural networks, etc. Finally, we summarize the research progress of PDE intelligent solutions, and outline the corresponding application scenarios and future research outlook.

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