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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.
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关键词:
- PINNs(基于物理信息神经网络) /
- 算子学习 /
- 计算力学 /
- AI for PDEs /
- 固体力学
Abstract: In recent years, deep learning has become ubiquitous and is empowering various fields. In particular, the combination of artificial intelligence and traditional science (AI for science, AI4Science) has attracted widespread attention. In the field of AI4Science, the use of artificial intelligence algorithms to solve partial differential equations (AI4PDEs) has become the focus of computational mechanics research. The core of AI4PDEs is to fuse data with equations and can solve almost any PDEs. Due to the advantages of AI4PDEs in data fusion, computational efficiency using AI4PDEs is usually increased by tens of thousands of times compared to traditional algorithms. Therefore, this article comprehensively reviews the research on AI4PDEs, summarizes the existing AI4PDEs algorithms and theories, discusses its application in solid mechanics, including forward and inverse problems, and outlines future research directions, especially the foundation model of computational mechanics. Existing algorithms of AI4PDEs include physics-informed neural networks (PINNs), deep energy methods (DEM), operator learning, and (physics-informed neural operator, PINO). AI4PDEs has numerous applications in scientific computing, and this paper focuses on application of AI4PDEs in the forward and inverse problems of solid mechanics. The forward problems include linear elasticity, elasto-plasticity, hyperelasticity, and fracture mechanics; while the inverse problems encompass the identification of material parameters, constitutive laws, defect recognition, and topology optimization. AI4PDEs represents a novel method of scientific simulation, which offers approximate solutions for specific problems by leveraging large datasets and then fine-tunes according to the specific physical equations, avoiding the need to start calculations from scratch as traditional algorithms do. Thus, AI4PDEs is a prototype for the foundation model of computational mechanics in the future, capable of significantly accelerating traditional numerical methods. We believe that utilizing artificial intelligence to empower scientific computing is not only a vital direction for the future of computation but also a dawn of humanity in scientific research, laying the foundation for mankind to reach new heights in scientific development.-
Key words:
- PINNs /
- operator learning /
- computational mechanics /
- AI for PDEs /
- solid mechanics
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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 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 
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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
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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
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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
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