Physics-informed neural networks (PINNs) have shown remarkable prospects in the solving the forward and inverse problems involving partial differential equations (PDEs). The method embeds PDEs into the neural network by calculating PDE loss at a series 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-weighted residual and propose volume-weighted 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 spatial distribution characteristics of collocation points in the residual 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 verify 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 3 times, and can reduce the relative error of conventional PINNs in solving the inverse problem by more than one order of magnitude.

翻译：物理信息神经网络（PINNs）在求解涉及偏微分方程（PDE）的正问题和逆问题中展现出显著前景。该方法通过在配置点集合上计算PDE损失将PDE嵌入神经网络，具有无需网格、自适应采样更便捷等优势。然而，在使用非均匀配置点求解PDE时，PINNs仍面临PDE残差收敛效率低下甚至失效的挑战。本文首先分析了非均匀配置点下PINNs中PDE损失的条件恶化问题。为解决该问题，我们定义了体积加权残差，并提出体积加权物理信息神经网络（VW-PINNs）。通过用配置点在计算域内占据的体积加权PDE残差，我们将配置点的空间分布特征显式嵌入残差评估中，确保了涉及非均匀配置点问题时PDE残差的快速充分收敛。基于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来计算配置点的体积。通过求解圆柱绕流和不同来流条件下NACA0012翼型绕流等传统PINNs难以收敛的正问题，验证了VW-PINNs的普适性；通过求解Burgers方程，验证了VW-PINNs能将现有自适应采样方法求解正问题的效率提升3倍，并将传统PINNs求解逆问题的相对误差降低超过一个数量级。