aiqbench analysis
This research introduces PVD-ONet, a novel operator learning framework designed to overcome the limitations of standard physics-informed neural networks when solving singularly perturbed partial differential equations. By integrating Prandtl's matching condition and Van Dyke's principle, the method achieves superior stability and accuracy in capturing fine-scale boundary layer structures without requiring retraining for new problem families. The approach also demonstrates capability in inverse problems, allowing inference of physical scaling exponents from sparse data, which is significant for scientific computing applications.
PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems
arXiv:2507.21437v3 Announce Type: replace Abstract: 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.
Abstract: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.
Machine Learning (cs.LG)
arXiv:2507.21437 [cs.LG]
(or arXiv:2507.21437v3 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2507.21437 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 236 words
