Two step training a single physics-informed neural network for solving Navier Stokes equations with various boundary conditions

Abstract

Physics-Informed Neural Networks (PINNs) are a popular scientific machine learning framework used to solve partial differential equations (PDEs). One of the common applications of PINNs is in solving fluid flow problems using the Navier–Stokes (NS) equations. The NS equations are a set of PDEs that describe the flow of a viscous fluid and have been extensively applied in manufacturing problems, such as modeling flow in injection molding or the flow of molten metal in additive manufacturing. Solving a single PINN with various boundary conditions requires training a unified model to predict the flow field for each specific boundary condition setup. This poses a challenge in training PINNs due to the limited number of samples that can be taken from the parametric space corresponding to various boundary conditions, often leading to poor-quality solutions. To address this, we propose a two-step solution to solve PINNs for the Navier–Stokes equations with various boundary conditions. The proposed method enables PINNs to learn effectively both from the domain and from parametric spaces. This two-step approach provides the model with a finer initial understanding of the domain space and then shifts to sampling from the parametric space to enhance its knowledge of the parametric variations. Numerical studies demonstrate the effectiveness of the proposed approach compared to direct training of PINNs. Increased knowledge about domain space provides the model with better learning of boundary conditions and lower PDE residuals. The proposed method uses the same computational requirements as direct training but provides better convergence. Furthermore, the ability to learn parametric boundary conditions enables PINNs to be applied to a variety of versatile applications.