A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations

Yuan, Lei; Ni, Yiqing; Deng, Xiangyun; Hao, Shuo

doi:10.1016/j.jcp.2022.111260

Cited by 120 publications

(42 citation statements)

References 50 publications

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“…PINN is also compared against other methods that can be applied to solve PDEs, like the one based on the Feynman-Kac theorem [19]. Finally, PINNs have also been extended to solve integrodifferential equations (IDEs) [128,193] or stochastic differential equations (SDEs) [189,195].…”

Section: What the Pinns Arementioning

confidence: 99%

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

et al. 2022

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Physics-Informed Neural Networks (PINN) are neural networks (NNs) that encode model equations, like Partial Differential Equations (PDE), as a component of the neural network itself. PINNs are nowadays used to solve PDEs, fractional equations, integral-differential equations, and stochastic PDEs. This novel methodology has arisen as a multi-task learning framework in which a NN must fit observed data while reducing a PDE residual. This article provides a comprehensive review of the literature on PINNs: while the primary goal of the study was to characterize these networks and their related advantages and disadvantages. The review also attempts to incorporate publications on a broader range of collocation-based physics informed neural networks, which stars form the vanilla PINN, as well as many other variants, such as physics-constrained neural networks (PCNN), variational hp-VPINN, and conservative PINN (CPINN). The study indicates that most research has focused on customizing the PINN through different activation functions, gradient optimization techniques, neural network structures, and loss function structures. Despite the wide range of applications for which PINNs have been used, by demonstrating their ability to be more feasible in some contexts than classical numerical techniques like Finite Element Method (FEM), advancements are still possible, most notably theoretical issues that remain unresolved.

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Section: What the Pinns Arementioning

confidence: 99%

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

et al. 2022

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“…Chen 和他的团队对一些经典的数学物理方程进行了研究，如 Sine-Gordon 方程、非线性薛定谔方程和导数非线性薛定谔方程，获得了这些方程的数据驱动的呼吸子解、怪波解和其他孤子解 [23][24][25][26][27][28] ；并且在 PINNs 模型的基础上，提出了一种两阶段物理信息神经网络 [29] ，对一类非线性物理方程进行了模拟，获得了这些方程的局域波解。Ling [30] 等人使用改进的 PINNs 模型对耦合非线性薛定谔方程进行模拟，并获得了数据驱动的向量孤子解。He 等人 [31] 使用 PINNs 模型对非线性薛定谔方程在扰动初始条件下的 Peregrine 孤子解进行了模拟。Yan 和 Wang 等人使用改进的 PINNs 模型研究了相关方程的数据驱动解和参数发现 [32][33][34][35] , 并通过改进的 PINNs 模型对 CH、DP 和 Novikov 等方程的 peakon 解和周期解进行了预测 [36] 。Bai 等人使用改进的 PINNs 模型求解了 Huxley 方程 [37] 。Wu 等人通过改进的 PINNs 模型预测了双折射光纤中矢量光孤子的动力学过程和模型参数 [38] 。Li 等人使用 PINNs 模型模拟了带有广义 PT 对称 Scarf-II 势的 NLS 方程 [39] ; 并且在 PINNs 模型的基础上提出了梯度优化物理信息神经网络模型 [40] 和混合训练物理信息神经网络模型 [41] ，这两种模型把模拟解的精度进一步提高。Dai 等人通过一种改进的 PINNs 模型模拟了包括非线性薛定谔方程、 KdV 方程和 mKdV 方程在内的孤子解 [42,43] 。 Yuan 等人通过 PINNs 模型解决了非线性积分微分方程的正逆问题 [44] 。Zeng 等人通过一种自适应神经网络模型模拟了高维偏微方程问题 [45] 。基于梯度均衡优化 [46] 的思想，我们提出了一种梯度优化的物理信息神经网络 (GOPINNs)。具体而言，通过对原始 PINNs 模型进行梯度统计，在模型训练期间平衡损失函数中不同项之间的相互作用，优化了梯度下降更新过程的方法，并改变其完全连接的前馈神经网络结构。这种改进的方法有两个主要动机：(1) 在模型训练期间，使用反向传播梯度统计 [47] 自动调整损失函数项系数，以平衡损失函数项之间的相互作用； (2) 该想法来源于最近频繁使用于 CV 和 NLP 研究的神经网络注意力机制 [48]…”

Section: 它可以有效地解决大域或多尺度解决方案。unclassified

Solving complex nonlinear problems based on gradient-optimized physics-informed neural networks

Tian¹,

Li²

2023

Acta Phys. Sin.

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In recent years, physics-informed neural networks (PINNs) have attracted more and more attention for their ability to quickly obtain high-precision data-driven solutions with only a small amount of data. However, although this model has good results in some nonlinear problems, it still has some shortcomings. For example, the unbalanced back-propagation gradient calculation results in the intense oscillation of the gradient value during the model training, which is easy to lead to the instability of the prediction accuracy. Based on this, we propose a gradient-optimized physics-informed neural networks (GOPINNs) model in this paper, which proposes a new neural network structure and balances the interaction between different terms in the loss function during model training through gradient statistics, so as to make the new proposed network structure more robust to gradient fluctuations. In this paper, taking Camassa-Holm(CH) equation and DNLS equation as examples, GOPINNs is used to simulate the peakon solution of CH equation, the rational wave solution of DNLS equation and the rogue wave solution of DNLS equation. The numerical results show that the GOPINNs can effectively smooth the gradient of the loss function in the calculation process, and obtain a higher precision solution than the original PINNs. In conclusion, our work provides new insights for optimizing the learning performance of neural networks, and saves more than one third of the time in simulating the complex CH equation and the DNLS equation, and improves the prediction accuracy by nearly ten times.

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“…Deep learning applications in MFL testing for pipeline defect quantification have received significant research attention [22][23][24]. This work was motivated by deep learning's ability to automatically learn from data representing features.…”

Section: Introductionmentioning

confidence: 99%

“…It is worthwhile noting that the vast majority of MFL studies with deep learning are conducted in a data-driven manner [27,28]. There is a risk of violating the physical knowledge underlying MFL testing and yielding nonsensical predictions [24]. This highlights the need to integrate physical knowledge with MFL data for defect quantification [29].…”

Section: Introductionmentioning

confidence: 99%

Magnetic flux leakage defect size estimation method based on physics-informed neural network

Xiong,

Liu,

Hou

et al. 2023

Phil. Trans. R. Soc. A.

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Magnetic flux leakage (MFL) is a magnetic method of non-destructive testing for in-pipe defect detection and sizing. Despite the fact that recent developments in machine learning have revolutionized disciplines like MFL defect size estimation, the most current methods for quantifying pipeline defects are primarily data-driven, which may violate the underlying physical knowledge. This paper proposes a physics-informed neural network-based method for MFL defect size estimation. The training process of neural network is guided by the MFL data and the physical constraints that is mathematically represented by the magnetic dipole model. We use synthetic MFL data produced by a virtual MFL testing of pipeline defects to validate the proposed method through a comparison to purely data-driven neural networks and support vector machines. The findings imply that the physics-informed strategy can both improve predictive accuracy and mitigate physical violations in MFL testing, providing us with a better knowledge of how neural networks perform in defect size estimation. This article is part of the theme issue ‘Physics-informed machine learning and its structural integrity applications (Part 2)’.

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A-PINN: Auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations

Cited by 120 publications

References 50 publications

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

Scientific Machine Learning Through Physics–Informed Neural Networks: Where we are and What’s Next

Solving complex nonlinear problems based on gradient-optimized physics-informed neural networks

Magnetic flux leakage defect size estimation method based on physics-informed neural network

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