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.

Recently, many variational models involving high order derivatives have been widely used in image processing, because they can reduce staircase effects during noise elimination. However, it is very challenging to construct efficient algorithms to obtain the minimizers of original high order functionals. In this paper, we propose a new linearized augmented Lagrangian method for Euler's elastica image denoising model. We detail the procedures of finding the saddle-points of the augmented Lagrangian functional. Instead of solving associated linear systems by FFT or linear iterative methods (e.g., the Gauss-Seidel method), we adopt a linearized strategy to get an iteration sequence so as to reduce computational cost. In addition, we give some simple complexity analysis for the proposed method. Experimental results with comparison to the previous method are supplied to demonstrate the efficiency of the proposed method, and indicate that such a linearized augmented Lagrangian method is more suitable to deal with large-sized images.