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.

In the continuous-time random walk model, the time-fractional operator usually expresses an infinite waiting time probability density. Different from that usual setting, this work considers the tempered time-fractional operator, which reflects a finite waiting time probability density. Firstly, we analyse the solution of a tempered benchmark problem, which shows a weak singularity near the initial time. The L1 scheme on graded mesh and the weighted shifted Grünwald–Letnikov formula with correction terms are adapted to deal with the non-smooth solution, in which we compare these two methods systematically in terms of the convergence and consumed CPU time. Furthermore, a fast calculation for the time tempered Caputo fractional derivative is developed based on a sum-of-exponentials approximation, which significantly reduces the running time. Moreover, the tempered operator is applied to the Bloch equation in nuclear magnetic resonance and a two-layered problem with composite material exhibiting distinct memory effects, for which both the analytical (or semi-analytical) and numerical solutions are derived using transform techniques and finite difference methods. Data-fitting results verify that the tempered time-fractional model is much effective to describe the MRI data. An important finding is that, compared with the fractional index, the tempered operator parameter could further accelerate the diffusion. The tempered model with two parameters $$\alpha $$α and $$\rho $$ρ are more flexible, which can avoid choosing a too small fractional index leading to low regularity and strong heterogeneity.

Wang C, et al. Korean Circ J. 2024 Sep;54:e104. https://doi.org/10.4070/kcj.2024.0219

Background Understanding how symptomatic intracranial atherosclerotic disease (sICAD) evolves and identifying high-risk lesion characteristics could improve risk stratification. Purpose To delineate the evolution patterns of plaque morphologic features and focal hemodynamics in sICAD and identify morphologic features and hemodynamics associated with plaque progression and recurrence of stroke and/or transient ischemic attack (TIA). Materials and Methods In this prospective, longitudinal study, participants with high-grade (60%-99% stenosis) sICAD confirmed at three-dimensional rotational angiography (3DRA) between August 2007 and December 2021 received guideline-based medical treatment, with follow-up evaluations at 1, 3, 6, and 12 months and every 6 months afterward. All participants underwent follow-up 3DRA at 1 year or upon stroke recurrence if earlier. The sICAD plaque morphologic features and focal hemodynamics were assessed and compared at baseline and follow-up 3DRA and using 3DRA-based computational fluid dynamics models. The primary outcome was a composite of relevant ischemic stroke or hard TIA recurrence and/or plaque progression in 3DRA at 1 year. Results Among 97 participants (median age, 62 years [IQR, 56-68 years]; 68 male), the primary outcome occurred in 26 (27%): 17 (18%) with relevant ischemic stroke and/or TIA recurrence and nine (9%) with plaque progression. Focal pressure and wall shear stress measures also changed over 1 year. Greater plaque thickness (adjusted odds ratio, 10.49 [95% CI: 2.67, 41.20]), smaller upstream plaque shoulder angulation (adjusted odds ratio, 0.97 [95% CI: 0.95, 1.00]), and maximum stenosis located distally (adjusted odds ratio, 3.81 [95% CI: 1.19, 12.18]) at baseline were independently associated with primary outcome. Higher and lower wall shear stress measures across the upstream and downstream plaque segments, respectively, at baseline were also associated with primary outcome. Conclusion Plaque morphologic features and focal hemodynamics evolved over time in participants with sICAD under medical treatment; thicker plaque, smaller upstream shoulder angulation, and skewed distal load were associated with lesion progression and recurrence of stroke and/or TIA. © The Author(s) 2026. Published by the Radiological Society of North America under a CC BY 4.0 license. Supplemental material is available for this article.