VW-PINNs:物理信息神经网络中 PDE 残差的体积加权法

IF 3.8 2区 工程技术 Q1 ENGINEERING, MECHANICAL Acta Mechanica Sinica Pub Date : 2024-07-30 DOI:10.1007/s10409-024-24140-x
Jiahao Song  (, ), Wenbo Cao  (, ), Fei Liao  (, ), Weiwei Zhang  (, )
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引用次数: 0

摘要

物理信息神经网络(PINNs)在解决涉及偏微分方程(PDEs)的正向和反向问题方面展现出了广阔的前景。该方法通过计算一组配位点的 PDE 损失,将 PDE 嵌入神经网络,具有无网格和自适应采样更方便等优点。然而,在使用非均匀配位点求解 PDE 时,PINN 仍然面临 PDE 残差收敛效率低下甚至失效的挑战。在这项工作中,我们首先分析了在非均匀配置点下 PINN 中 PDE 损失的非条件性。为了解决这个问题,我们定义了体积加权残差,并提出了体积加权物理信息神经网络(VW-PINNs)。通过对 PDE 残差进行计算域内搭配点所占体积的加权,我们在损失评估中明确嵌入了搭配点的分布特征。对于涉及非均匀配点的问题,保证了 PDE 残差的快速充分收敛。考虑到 VW-PINNs 的无网格特性,我们还开发了一种基于核密度估计的体积近似算法,用于计算拼合点的体积。我们通过求解圆柱体和 NACA0012 机翼在不同流入条件下的正向流动问题验证了 VW-PINNs 的通用性,而传统的 PINNs 在这些条件下是失效的。通过求解布尔格斯方程,我们验证了 VW-PINN 在求解正向问题时可将现有自适应采样方法的效率提高三倍,并可将传统 PINN 在求解反向问题时的相对 L2 误差降低一个数量级以上。
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VW-PINNs: A volume weighting method for PDE residuals in physics-informed neural networks

Physics-informed neural networks (PINNs) have shown remarkable prospects in solving the forward and inverse problems involving partial differential equations (PDEs). The method embeds PDEs into the neural network by calculating the PDE loss at a set of collocation points, providing advantages such as meshfree and more convenient adaptive sampling. However, when solving PDEs using nonuniform collocation points, PINNs still face challenge regarding inefficient convergence of PDE residuals or even failure. In this work, we first analyze the ill-conditioning of the PDE loss in PINNs under nonuniform collocation points. To address the issue, we define volume weighting residual and propose volume weighting physics-informed neural networks (VW-PINNs). Through weighting the PDE residuals by the volume that the collocation points occupy within the computational domain, we embed explicitly the distribution characteristics of collocation points in the loss evaluation. The fast and sufficient convergence of the PDE residuals for the problems involving nonuniform collocation points is guaranteed. Considering the meshfree characteristics of VW-PINNs, we also develop a volume approximation algorithm based on kernel density estimation to calculate the volume of the collocation points. We validate the universality of VW-PINNs by solving the forward problems involving flow over a circular cylinder and flow over the NACA0012 airfoil under different inflow conditions, where conventional PINNs fail. By solving the Burgers’ equation, we verify that VW-PINNs can enhance the efficiency of existing the adaptive sampling method in solving the forward problem by three times, and can reduce the relative L2 error of conventional PINNs in solving the inverse problem by more than one order of magnitude.

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来源期刊
Acta Mechanica Sinica
Acta Mechanica Sinica 物理-工程:机械
CiteScore
5.60
自引率
20.00%
发文量
1807
审稿时长
4 months
期刊介绍: Acta Mechanica Sinica, sponsored by the Chinese Society of Theoretical and Applied Mechanics, promotes scientific exchanges and collaboration among Chinese scientists in China and abroad. It features high quality, original papers in all aspects of mechanics and mechanical sciences. Not only does the journal explore the classical subdivisions of theoretical and applied mechanics such as solid and fluid mechanics, it also explores recently emerging areas such as biomechanics and nanomechanics. In addition, the journal investigates analytical, computational, and experimental progresses in all areas of mechanics. Lastly, it encourages research in interdisciplinary subjects, serving as a bridge between mechanics and other branches of engineering and the sciences. In addition to research papers, Acta Mechanica Sinica publishes reviews, notes, experimental techniques, scientific events, and other special topics of interest. Related subjects » Classical Continuum Physics - Computational Intelligence and Complexity - Mechanics
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