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

Physics-informed neural networks and physics-informed DeepONet excel at solving partial differential equations but often struggle with singularly perturbed problems. To address this limitation, we propose two novel data-free frameworks: the Prandtl--Van Dyke neural network(PVD-Net) and its operator-learning extension, the Prandtl--Van Dyke Deep Operator Network(PVD-ONet). Each framework provides leading-order and high-order versions to meet different modeling requirements. The leading-order PVD-Net integrates Prandtl's matching condition for improved stability, while the high-order PVD-Net employs Van Dyke's matching principle to capture fine-scale boundary layer structures with higher accuracy. Extending PVD-Net to operator learning, PVD-ONet enables efficient prediction for families of boundary layer problems without retraining. Numerical experiments on representative one- and two-dimensional problems demonstrate that the proposed methods outperform existing baselines. Beyond forward prediction, the framework also supports inverse problems by inferring the scaling exponent governing boundary layer thickness from sparse observations, highlighting its potential for practical applications.