Physics-informed neural networks with domain decomposition for the incompressible Navier–Stokes equations

Physics-informed neural network (PINN) has emerged as a promising approach for solving differential equations in recent years. However, their application to large-scale complex problems has faced challenges regarding accuracy and efficiency. To address these limitations, domain decomposition has gained popularity as an effective strategy. This paper studies a domain decomposition PINN method for solving incompressible Navier–Stokes equations. We assess the method's predicted accuracy, convergence, and the impact of different strategies on performance. In the domain decomposition PINN method, individual PINN is employed for each subdomain to compute local solutions, which are seamlessly connected by enforcing additional continuity conditions at the interfaces. To improve the method's performance, we investigate various continuity conditions at the interfaces and analyze their influence on the predictive accuracy and interface continuity. Furthermore, we introduce two approaches: the dynamic weight method and a novel neural network architecture incorporating attention mechanisms, both aimed at mitigating gradient pathologies commonly encountered in PINN methods. To demonstrate the effectiveness of the proposed method, we apply it to a range of forward and inverse problems involving diverse incompressible Navier–Stokes flow scenarios. This includes solving benchmark problems such as the two-dimensional (2D) Kovasznay flow, the three-dimensional (3D) Beltrami flow, the 2D lid-driven cavity flow, and the 2D cylinder wake. Additionally, we conduct 3D blood flow simulations for synthetic flow geometries and real blood vessels. The experimental results demonstrate the capability and versatility of the domain decomposition PINN method in accurately solving incompressible Navier–Stokes flow problems.