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. (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(Physics-informed neural networks, PINNs)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(Partial differential equations, PDEs)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)PDE(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic),(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), PINNs(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)PINNs(sic)PDE(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(Volume weighting physics-informed neural networks, VW-PINNs). (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)PDE(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic), (sic)(sic)(sic)VW-PINNs(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)(sic)(sic)PINNs(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic),(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)VW-PINNs(sic)(sic)(sic)(sic). (sic)(sic)(sic)(sic)Burgers'(sic)(sic), (sic)(sic)(sic)(sic)(sic)VW-PINNs(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic), (sic)(sic)(sic)PINNs(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)L2(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic)(sic).