Grid residual adaptive resampling for physics-informed neural networks to solve incompressible Navier–Stokes equations

This study aims to tackle the challenges of nonlinearity and coupling that Physics-Informed Neural Networks (PINNs) encounter when solving the incompressible Navier–Stokes equations. The research core is centered around optimizing the sampling strategy of residual collocation points to enhance PINNs’ capability of capturing the key features of the Navier–Stokes solutions. To this end, we propose a Grid Residual Adaptive Resampling (GRAR) strategy to improve the accuracy of PINNs in solving the Navier–Stokes equations. The GRAR-PINN method selectively samples residual collocation points based on grid residual values, allowing it to more effectively capture the essential characteristics of the Navier–Stokes solutions. To validate the effectiveness of this approach, we carried out numerical experiments using two-dimensional steady Kovasznay flow and two-dimensional triangular cylinder flow. The results show that the GRAR-PINN method can significantly enhance the accuracy of PINNs in solving the Navier–Stokes equations with relatively fewer residual collocation points through continuous resampling. This method not only alleviates the problem of repeatedly adjusting the number of residual collocation points but also addresses the challenge of capturing the key features of the Navier–Stokes solutions using PINNs.