PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems

PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems Tiantian Sun, Jian Zu PVD-Net: a physics-informed framework for singularly perturbed problems. Van Dyke matching principle is introduced to enhance prediction accuracy. Numerical results show our methods exceed baseline performance. Abstract Physics-informed neural networks and Physics-informed DeepONet excel in solving partial differential equations; however, they often fail to converge for singularly perturbed problems. To address this, we propose two novel frameworks, Prandtl-Van Dyke neural network (PVD-Net) and its operator learning extension Prandtl-Van Dyke Deep Operator Network (PVD-ONet), which rely solely on governing equations without data. To address varying task-specific requirements, both PVD-Net and PVD-ONet are developed in two distinct versions, tailored respectively for stability-focused and high-accuracy modeling. The leading-order PVD-Net adopts a two-network architecture combined with Prandtl's matching condition, targeting stability-prioritized scenarios. The high-order PVD-Net employs a five-network design with Van Dyke's matching principle to capture fine-scale boundary layer structures, making it ideal for high-accuracy scenarios. Numerical experiments on various models show that our proposed methods consistently outperform existing baselines under various error metrics, thereby offering a powerful new approach for multi-scale problems. Keywords: Singularly perturbed problem; Matched asymptotic expansions; Machine learning; Operator learning; Multi-scale problem1. Introduction Singular perturbation problems are prevalent in various scientific and engineering disciplines, including fluid mechanics [1, 2], aerodynamics [3, 4, 5], solid mechanics [6, 7, 8], and biological transport [9]. For example, in aerodynamics, singular perturbation methods are used to analyze rapid pressure and velocity changes around a thin airfoil.