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

arXiv: https://arxiv.org/abs/2507.21437 · HTML: https://mlsanity.com/paper/2507.21437/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 51 references that are in the map.
- First posted on arXiv 29 Jul 2025.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [Physics-informed deep learning for incompressible laminar flows](https://mlsanity.com/paper/2002.10558/) (2020)

## Built on by (most cited first, 2 of 2)

- [GlueNN: gluing patchwise analytic solutions with neural networks](https://mlsanity.com/paper/2601.05889/) (2026)
- [Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High P'eclet Entrance Transport](https://mlsanity.com/paper/2608.19658/) (2026)

## Abstract (arXiv)

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.