Physics-informed neural network combined with characteristic-based split for solving 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). Our method separates the variables, so it is no need to consider the weight coefficients between different loss functions. As not all partial derivatives are involved in the gradient backpropagation, our method can save half of the memory occupation without losing accuracy, and resulting in a significant reduction in computation time compared to traditional PINN. We solving the progress of the dispersion of hot water under known flow fields. 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 to estimate the diffusion coefficient of hot water flow, which requires information from the field at only two different times. 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.