Adaptive multi-scale gradient-enhanced sampling physics-informed neural network for incompressible flow

Physics-Informed Neural Network (PINN) has emerged as a new paradigm for flow field prediction due to its ability to combine data-driven fitting and physical consistency. However, traditional PINN tends to fall into local optima when handling incompressible flows, and conventional sampling methods struggle to cover regions with high residuals or gradients, leading to reduced prediction accuracy for complex flows such as boundary layers and vortices. To address these issues, this paper proposes an Adaptive Multi-Scale Gradient-Enhanced Sampling PINN method, referred to as AMGS-PINN. The core innovation of this method lies in the development of an Adaptive Multi-Scale Gradient Regularization. This method utilizes residual gradient information at different scales to guide the network optimization process, thereby suppressing training instability and enhancing the model's generalization ability. Additionally, by combining clustering and adaptive sampling techniques, the method selects fine-scale points and coarse-scale points with representative physical and geometric characteristics during the adaptive sampling process, improving the representativeness of the sample distribution and the global stability of training. The effectiveness of AMGS-PINN is validated through four typical flow cases: Kovasznay flow, lid-driven cavity flow, Taylor–Green vortex, and unsteady cylinder flow. The results demonstrate that AMGS-PINN significantly improves computational accuracy and exhibits stronger stability compared to traditional PINN.