Physics-informed neural network combined with characteristic-based split for solving forward and inverse problems involving Navier–Stokes equations

This paper presents a novel approach for solving the shallow-water transport equation using a physics-informed neural network (PINN) combined with characteristic-based split (CBS). Simulation of the tide in East Coast of China is performed to verify the applicability of the present method in a practical problem. Furthermore, we propose a boundary condition that accounts for the second-order partial derivative term, which is more appropriate for solving the diffusion equation with open domains than the commonly used assumption of zero boundary values. Our numerical results demonstrate that this boundary condition leads to improved convergence of the network. In addition, we introduce a parameter estimation method that requires information from the field at only two different times, yet yields accurate parameter estimates. We observe that excessive participation of variables in gradient backpropagation can lead to neural networks getting trapped in local optima. We use PINN combined with CBS method to solve 3-D incompressible flow. As the number of variables involved in gradient backpropagation increases, the accuracy of the solution decreases, which can partially support our viewpoint. The source codes for the numerical examples in this work are available at https://github.com/double110/PINN-cbs-.git.