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AI for PDEs在固体力学领域的研究进展

王一铮 庄晓莹 TimonRabczcuk 刘应华

王一铮, 庄晓莹, Timon Rabczcuk, 刘应华. AI for PDEs在固体力学领域的研究进展. 力学进展, 2024, 54(4): 1-57 doi: 10.6052/1000-0992-24-016
引用本文: 王一铮, 庄晓莹, Timon Rabczcuk, 刘应华. AI for PDEs在固体力学领域的研究进展. 力学进展, 2024, 54(4): 1-57 doi: 10.6052/1000-0992-24-016
Wang Y Z, Zhuang X Y, Timon R, Liu Y H. AI for PDEs in solid mechanics: A review. Advances in Mechanics, 2024, 54(4): 1-57 doi: 10.6052/1000-0992-24-016
Citation: Wang Y Z, Zhuang X Y, Timon R, Liu Y H. AI for PDEs in solid mechanics: A review. Advances in Mechanics, 2024, 54(4): 1-57 doi: 10.6052/1000-0992-24-016

AI for PDEs在固体力学领域的研究进展

doi: 10.6052/1000-0992-24-016 cstr: 32046.14.1000-0992-24-016
基金项目: 国家自然科学基金 (12332005)资助, Timon Rabczuk教授的指导和讨论
详细信息
    作者简介:

    刘应华, 清华大学航天航空学院长聘教授/博导. 获国家杰出青年科学基金, 新世纪百千万人才工程国家级人选, 国家高层次创新人才计划. 长期从事复杂服役环境下破坏力学与结构完整性研究, 在塑性本构理论、蠕变断裂力学及结构完整性评估领域做出了系统性的创新成果, 形成了较为完整的理论和应用方法体系, 在国内外相关领域产生了重要的学术影响. 担任10多个国内外重要学术机构或组织的负责人、专家或委员, 以及6份国内核心期刊和5份国际SCI期刊的编委. 发表高水平学术论文300多篇, 其中SCI收录200余篇, 出版专著4部. 获国家发明专利25项、软件著作权18项. 负责制定国家标准2项和政府技术规范3项. 获2011年和2005年国家科技进步二等奖以及部级一等奖4 项, 2012年美国机械工程师学会ASME PVPD颁发的J. M. Chalmers奖, 2016年亚太工程塑性力学奖

    通讯作者:

    yhliu@mail.tsinghua.edu.cn

AI for PDEs in solid mechanics: A review

More Information
  • 摘要: 近几年, 深度学习无所不在, 在给各个领域赋能, 尤其是人工智能与传统科学的结合 (AI for science, AI4Science) 引发广泛关注. 在AI4Science领域, 利用人工智能算法求解PDEs (AI4PDEs) 成为计算力学研究的焦点. AI4PDEs的核心是将数据与方程相融合, 并且几乎可以求解任何偏微分方程问题, 由于其融合数据的优势, 相较于传统算法, 其计算效率通常提升数万倍. 因此, 本文全面综述了AI4PDEs的研究, 总结了现有AI4PDEs算法、理论, 并讨论了其在固体力学中的应用, 包括正问题和反问题, 展望了未来研究方向, 尤其是必然会出现的计算力学大模型. 现有AI4PDEs算法包括基于物理信息神经网络 (physics-informed neural network, PINNs)、深度能量法 (deep energy methods, DEM)、算子学习 (operator learning), 以及基于物理神经网络算子 (physics-informed neural operator, PINO). AI4PDEs在科学计算中有许多应用, 本文聚焦于固体力学, 正问题包括线弹性、弹塑性, 超弹性、以及断裂力学; 反问题包括材料参数, 本构, 缺陷的识别, 以及拓朴优化. AI4PDEs代表了一种全新的科学模拟方法, 通过利用大量数据在特定问题上提供近似解, 然后根据具体的物理方程进行微调, 避免了像传统算法那样从头开始计算, 因此AI4PDEs是未来计算力学大模型的雏形, 能够大大加速传统数值算法. 我们相信, 利用人工智能助力科学计算不仅仅是计算领域的未来重要方向, 同时也是科学研究的人类曙光, 为人类迈向科学发展的新高度奠定了基础.

     

  • 图  1  AI4PDEs在AI4Science中扮演的角色, 以及AI4PDE的介绍, AI4PDEs的核心是利用人工智能算法帮助求解PDEs, AI4PDEs大体分为正问题和反问题 (Zhang et al. 2023)

    图  2  AI4PDEs主要的方法: 基于物理神经网络, 算子学习, 基于物理神经算子. 基于物理神经网络的核心是近似函数用神经网络代替. 算子学习的核心是基于数据驱动算法学习PDEs族的映射关系. 基于物理神经算子的核心是将算子学习和物理方程进行结合

    图  3  AI for PDEs方法: PINNs强形式示意图 (Cuomo et al. 2022)

    图  4  AI for PDEs方法: PINNs能量原理. (a) 基于最小势能原理的深度能量法DEM (Nguyen-Thanh et al. 2020, Samaniego et al. 2020, Wang Y et al. 2022), (b) 基于最小余能原理的深度余能法(Wang et al. 2023)

    图  5  AI for PDEs方法: 算子学习示意图. (a) 神经算子算法(Kovachki et al. 2023), (b) DeepONet (Lu et al. 2021a), (c) FNO (Li et al. 2020a)

    图  6  AI for PDEs方法: 基于物理神经算子示意图. (a) 利用物理方程对算子学习的初解进行微调(Li Z et al. 2021), (b) 利用数据对近似物理方程的低精度解进行微调(Chakraborty 2021)

    图  7  PINNs误差和有限元误差的比较: 我们从函数空间的角度将误差分为, 优化, 积分和近似误差

    图  8  PINNs在线弹性力学中的算法上的应用. (a) 强形式: 线弹性动力学(Rao et al. 2021), (b) 强形式和能量形式: Kirchhoff板壳(Li W et al. 2021 ), (c) 反形式: 边界元(Sun et al. 2023a)

    图  9  算子学习在线弹性力学中的均匀化中的应用. (a) 预测等效弹性模量(Wang et al. 2024), (b) 多尺度RVE代表性体积元(Liu et al. 2024a)

    图  10  PINNs和算子学习在弹塑性力学中的应用. (a) PINNs强形式(Abueidda et al. 2021)和能量形式(He et al. 2023a), (b) 算子学习Geo-FNO (Li et al. 2023)

    图  11  PINNs能量形式在超弹性力学中的应用. (a) PINNs能量形式(Fuhg & Bouklas 2022, Nguyen-Thanh et al. 2020), (b) PINNs能量子域形式(Wang Y et al. 2022)

    图  12  基于物理信息神经算子在断裂力学相场法中的应用. (a) PINNs能量形式(Goswami et al. 2020), (b) PINNs能量形式和算子结合(Goswami et al. 2022 )

    图  13  PINNs在弹性模量和泊松比的识别的应用. (a) 识别均匀各项同性弹塑形材料的弹性模量和泊松比, 以及屈服应力(Haghighat et al. 2021), (b) 识别非均匀各项同性材料的弹性模量场(Chen & Gu 2021), (c) 识别非均匀各项同性材料的热导率(Liu et al. 2024b)

    图  14  AI for PDEs在本构方程识别的应用. (a) 神经网络拟合应变和应力的超弹性关系(Li & Chen 2022), (b) 提前给定超弹性本构的形式, 拟合前面的参数(Flaschel et al. 2021)

    图  15  PINNs能量形式在拓扑优化的应用. (a) 密度场传统方法迭代(He et al. 2023c), (b) 神经网络拟合密度场(Jeong et al. 2023b)

    图  16  AI for PDEs在缺陷识别的应用. (a) PINNs识别缺陷(Zhang et al. 2022), (b) 数据驱动识别缺陷(Sun et al. 2023b)

    表  1  PINN强形式算法研究现状

    文献 方法 试函数分片 权函数分片 试函数类型 权函数类型 本质边界施加方式
    Raissi 等 (2019) PINNs 否 (全局) 否 (全局) 全连接 Delta 罚函数
    Sirignano和Spiliopoulos (2018) DGM 否 (全局) 否 (全局) 全连接 最小二乘 罚函数
    Kharazmi 等 (2019) VPINNs 否 (全局) 否 (全局) 全连接 混合权函数 罚函数
    Dwivedi和Srinivasan (2020) PIELM 是 (子域) 否 (全局) ELM (Huang et al. 2006) Delta 罚函数
    Jagtap 等 (2020c) cPINN 是 (子域) 否 (全局) 全连接 Delta 罚函数
    Jagtap和Karniadakis (2020) XPINN 是 (子域) 否 (全局) 全连接 Delta 罚函数
    Kharazmi 等 (2021) hp-VPINNs 否 (全局) 是 (子域) 全连接 正交多项式 罚函数
    Gao 等 (2021) PhyGeoNet 否 (全局) 否 (全局) 卷积网络 Delta 强制施加
    Gao 等 (2022) PIGCN 否 (全局) 否 (全局) 图卷积 有限元试函数空间 强制施加
    Ramabathiran和Ramachandran (2021) SPINN 否 (全局) 否 (全局) 径向基网络 有限元试函数空间 罚函数
    Sun 等 (2023a) BINN 否 (全局) 否 (全局) 全连接 微分方程基本解 边界积分项
    下载: 导出CSV

    表  2  PINN能量形式研究现状

    文献 方法 试函数分片 试函数类型 本质边界施加方式 能量原理
    (Yu 2018) Deep Ritz 否 (全局) 全连接 罚函数 变分原理
    (Samaniego et al. 2020) DEM 否 (全局) 全连接 距离, 边界, 广义网络 最小势能
    (Sheng & Yang 2021) PFNN 否 (全局) 全连接 距离, 边界, 广义网络 变分原理
    (Fuhg & Bouklas 2022) mDEM 否 (全局) 全连接 距离, 边界, 广义网络 混合形式
    (Nguyen-Thanh et al. 2021) P-DEM 否 (全局) 全连接 等参元 最小势能
    (Wang Y et al. 2022) CENN 是 (子域) 全连接 距离, 边界, 广义网络 最小势能
    (He et al. 2023b) GCN-DEM 否 (全局) 图卷积 距离, 边界, 广义网络 最小势能
    (Wang et al. 2023) DCEM 否 (全局) 全连接 距离, 边界, 广义网络 最小余能
    下载: 导出CSV

    表  3  PINNs权重选取研究现状

    文献方法简要描述
    (Wang et al. 2021a)通过比较不同成分的损失函数的梯度来选取权重
    (Wang S et al. 2022)利用NTK理论选取权重
    (Wang et al. 2021c)利用NTK理论识别出神经网络的频率倾向
    (Liu & Wang 2021)将PINNs损失函数修改成鞍点问题
    (Xu et al. 2023)利用最大似然估计修改损失函数
    下载: 导出CSV

    表  4  AI for PDEs收敛性证明

    文献 证明简要描述
    (Cohen et al. 2016, Cybenko 1989, Hornik et al. 1989, Pinkus 1999) 证明神经网络强大的拟合能力
    (Shin et al. 2020) 椭圆的二阶线性微分方程以及抛物线PDE, PINNs收敛性证明
    (Mishra & Molinaro 2022) PINNs反问题近似性证明
    (Psaros et al. 2023) PINNs不确定性估计
    (Kovachki et al. 2023) 神经算子近似任意连续算子的证明
    (Kovachki et al. 2021) 神经算子FNO近似任意连续算子的证明
    (Chen & Chen 1995) DeepONet理论支撑: 神经网络近似任意连续算子
    (Lanthaler et al. 2022) DeepONet近似任意连续算子的证明
    (De Ryck & Mishra 2022) PINO可以近似任意的连续函数或者任意的连续算子
    下载: 导出CSV
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