VW-PINNs：物理信息神经网络中偏微分方程残差的体积加权方法
VW-PINNs: A volume weighting method for PDE residuals in physics-informed neural networks
AI总结：
本文针对物理信息神经网络在非均匀配点下PDE残差收敛困难的问题，提出体积加权方法VW-PINNs，通过配点体积加权残差并基于核密度估计计算体积，显著提升了正逆问题求解效率与精度。
摘要：
物理信息神经网络（PINNs）在求解涉及偏微分方程（PDEs）的正问题和逆问题方面展现出显著前景。该方法通过在一系列配点上计算PDE损失，将PDE嵌入神经网络，具有无网格化和更便于自适应采样等优点。然而，当使用非均匀配点求解PDE时，PINNs仍面临PDE残差收敛效率低下甚至失败的问题。本文首先分析了非均匀配点下PINNs中PDE损失的病态性。为解决该问题，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）。通过用配点在计算域内所占体积对PDE残差进行加权，我们在残差评估中显式嵌入了配点的空间分布特征，从而保证了涉及非均匀配点问题的PDE残差快速且充分的收敛。考虑到VW-PINNs的无网格化特性，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点的体积。我们通过求解不同入流条件下绕圆柱流动和绕NACA0012翼型流动的正问题验证了VW-PINNs的普适性，而传统PINNs在这些问题上失效；通过求解Burgers方程，我们验证了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，并能将传统PINNs求解逆问题的相对误差降低一个数量级以上。
英文摘要：
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.
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