An improved physics-informed neural network with adaptive weighting and mixed differentiation for solving the incompressible Navier–Stokes equations

Physics-informed neural networks (PINNs) have been an effective tool for approximating the mapping between points in the spatio-temporal domain and solutions of partial differential equations (PDEs). However, there are still some challenges in dealing with the nonlinear characteristics and complexity of the Navier–Stokes (N–S) equations. In this paper, the improved adaptive weighting PINNs based on the Gaussian likelihood estimation are applied to solve the N–S equations. The weights of the different loss items are allocated adaptively by the maximum likelihood estimation. The improved network structure has been designed with considering both the global and local information, making it easier to capture the part of PDEs solution with drastic changes. A combinational method of the numerical differentiation (ND) and the automatic differentiation (AD) is proposed to compute the differential operators, with the improved computational efficiency. The derivative operation of the convection and pressure-gradient terms was carried out using the combined method in solving the incompressible N–S equations. The results show that the effectiveness and training efficiency of this method are better than PINNs.