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VW-PINNs: A volume weighting method for PDE residuals in physics-informed neural networks

VW-PINNs: 物理信息神经网络中PDE残差的体积加权方法

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Abstract

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.

摘要

物理信息神经网络(Physics-informed neural networks, PINNs)已经在偏微分方程(Partial differential equations, PDEs)正反问题求解领域展示出广阔前景. 该方法通过在一组配置点上计算PDE损失实现了偏微分方程在神经网络中的嵌入, 具有无网格、易于自适应采样等优势. 然而, 当使用非均匀配置点求解方程时, PINNs面临方程残差收敛效率低甚至求解失败的问题. 本研究首先分析了非均匀配置点下PINNs中PDE损失函数的病态. 然后针对性地定义了体积加权残差的概念, 并提出了体积加权的物理信息神经网络(Volume weighting physics-informed neural networks, VW-PINNs). 通过使用配置点在求解区域占据的体积对方程残差进行加权, 配置点的空间分布特征被显式地引入PDE损失函数中, 保证了方程残差快速充分的收敛. 同时, 考虑到VW-PINNs的无网格特性, 我们还基于核函数估计发展了近似算法来计算配置点的体积. 我们通过求解PINNs失效的不同来流条件下的圆柱、翼型绕流正问题验证了VW-PINNs的普适性. 通过求解Burgers’方程, 我们验证了VW-PINNs能够将现有自适应采样方法在正问题上的收敛速度提升三倍, 能够将PINNs在反问题上的相对L2误差降低一个数量级以上.

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 92152301), the National Key Research and Development Program of China (Grant No. 2022YFB4300200), and the Shaanxi Provincial Key Research and Development Program (Grant No. 2023-ZDLGY-27).

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Authors and Affiliations

  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, China

    Jiahao Song  (宋家豪), Wenbo Cao  (曹文博), Fei Liao  (廖飞) & Weiwei Zhang  (张伟伟)

  2. International Joint Institute of Artificial Intelligence on Fluid Mechanics, Northwestern Polytechnical University, Xi’an, 710072, China

    Jiahao Song  (宋家豪), Wenbo Cao  (曹文博) & Weiwei Zhang  (张伟伟)

  3. National Key Laboratory of Aircraft Configuration Design, Xi’an, 710072, China

    Weiwei Zhang  (张伟伟)

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  1. Jiahao Song  (宋家豪)
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Contributions

Author contributions Jiahao Song: Data curation, Formal analysis, Investigation, Methodology, Software, Validation, Visualization, Writing – original draft. Wenbo Cao: Formal analysis, Investigation, Software, Writing – review & editing. Fei Liao: Methodology, Supervision. Weiwei Zhang: Conceptualization, Funding acquisition, Supervision, Writing – review & editing.

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Correspondence to Weiwei Zhang  (张伟伟).

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Song, J., Cao, W., Liao, F. et al. VW-PINNs: A volume weighting method for PDE residuals in physics-informed neural networks. Acta Mech. Sin. 41, 324140 (2025). https://doi.org/10.1007/s10409-024-24140-x

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  • DOI: https://doi.org/10.1007/s10409-024-24140-x

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