cs.LGJul 29, 2025

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

Authors: Tiantian Sun, Jian Zu

Organizations: School of Mathematics and Statistics, Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun 130024, P.R. China

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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.

Figures & tables

MethodTrainingNew InstanceLeading-OrderHigh-OrderSmooth
 SpeedInferenceApproximationApproximationSolution
BL-PINNsFastNoYesNoNo
MSM-NNFastNoYesNoYes
PVD-Net (Ours)FastNoYes (Stable, Simple)Yes (High Precision)Yes
PVD-ONet (Ours)SlowYesYes (Stable, Simple)Yes (High Precision)Yes
Table 1: Comparison of neural network methods for singular perturbation problems.
Figure 1: Schematic diagram of the boundary layer scale transformation. The diagram illustrates the evolution of the boundary layer’s scale. The domain is decomposed into two parts: a boundary layer region Ωi\Omega_{i} and an outer region Ωo\Omega_{o} . By introducing the transformation ξ=x−x0ε\xi=\frac{x-x_{0}}{\varepsilon} , the boundary layer region Ωi\Omega_{i} is mapped to an amplified region Ωi′\Omega_{i}^{\prime} in the stretched coordinate.
Figure 2: Network architecture of PVD-Net. The leading-order version uses outer and inner networks with Prandtl matching. The high-order version employs five networks to construct leading- and first-order approximations through Van Dyke matching, with one of the first-order correction networks naturally reducing to the leading-order formulation. Both versions use composite solutions as the final output to ensure smooth transitions.
Figure 3: Architecture of the physics-informed DeepONet. The branch network encodes the input function vv sampled at fixed sensor locations x1,x2,…,xM{x_{1},x_{2},\ldots,x_{M}} , while the trunk network encodes the coordinate ζ\zeta . Their outputs are combined through a dot product to approximate the solution operator. Automatic differentiation evaluates the PDE residuals and enforces the physical constraints.
ItemSetting
OptimizerAdam
Learning rate10−410^{-4}
Training epochs100,000
InitializationKaiming (He)
ActivationSiLU
Model selectionBest checkpoint
Table 2: Training hyperparameter settings for all methods.
Figure 4: (Left): Prediction of the leading-order PVD-Net. (Right): Prediction of the high-order PVD-Net. Both versions accurately capture the boundary layer solution with smooth transitions. The high-order PVD-Net further improves accuracy over the leading-order version, particularly in the boundary layer and transition regions.
MethodGlobal ErrorsInner Region ErrorsJunction PointTraining
 Relative L2L^{2}L∞L^{\infty}Relative L2L^{2}L∞L^{\infty}ErrorTime (s)
BL-PINNs [ 3 ]1.50×10−31.50\times 10^{-3}2.27×10−22.27\times 10^{-2}1.50×10−31.50\times 10^{-3}2.27×10−22.27\times 10^{-2}2.21×10−22.21\times 10^{-2}655
MSM-NN [ 51 ]5.54×10−45.54\times 10^{-4}8.94×10−38.94\times 10^{-3}5.55×10−45.55\times 10^{-4}8.94×10−38.94\times 10^{-3}3.03×10−43.03\times 10^{-4}680
Leading-order PVD-Net9.43×10−49.43\times 10^{-4}5.39×10−35.39\times 10^{-3}9.44×10−49.44\times 10^{-4}5.39×10−35.39\times 10^{-3}5.20×10−35.20\times 10^{-3}766
High-order PVD-Net1.44×10−4\mathbf{1.44\times 10^{-4}}1.90×10−3\mathbf{1.90\times 10^{-3}}1.44×10−4\mathbf{1.44\times 10^{-4}}1.90×10−3\mathbf{1.90\times 10^{-3}}1.02×10−4\mathbf{1.02\times 10^{-4}}1,380
Table 3: Comparison of prediction errors and training time for PVD-Net on the constant-coefficient problem. The bolded values indicate the lowest errors among all methods.
Figure 5: Sensitivity to the truncation parameter ξ0\xi_{0} for leading-order PVD-Net. (Left): reconstructed solutions for ξ0=5,10,20,30\xi_{0}=5,10,20,30 . (Right): zoomed-in views near the boundary layer. A too small truncation ( ξ0=5\xi_{0}=5 ) leads to visible errors, while larger values yield nearly identical and accurate predictions.
Figure 6: (Left): Predicted results of Leading-order PVD-ONet. (Right): Predicted results of High-order PVD-ONet. We aim to learn the operator G:(a,b)↦uG:(a,b)\mapsto u .
MethodGlobal ErrorsInner Region ErrorsTraining
 Relative L2L^{2}L∞L^{\infty}Relative L2L^{2}L∞L^{\infty}Time (s)
PI-DeepONet [ 47 ]8.71×10−18.71\times 10^{-1}4.80×1004.80\times 10^{0}8.71×10−18.71\times 10^{-1}4.80×1004.80\times 10^{0}6,817
Leading-order PVD-ONet1.01×10−31.01\times 10^{-3}5.83×10−35.83\times 10^{-3}1.01×10−31.01\times 10^{-3}5.83×10−35.83\times 10^{-3}18,495
High-order PVD-ONet1.5×10−4\mathbf{1.5\times 10^{-4}}1.82×10−3\mathbf{1.82\times 10^{-3}}1.5×10−4\mathbf{1.5\times 10^{-4}}1.82×10−3\mathbf{1.82\times 10^{-3}}33,656
Table 4: Comparison of mean prediction errors for PVD-ONet on the constant-coefficient problem. Bold indicates the lowest error among all methods.
Figure 7: Prediction results of the PVD-Net for the internal layer problem. (Left): Predicted results of Leading-order PVD-Net. (Right): Predicted results of High-order PVD-Net. A closer examination through localized zoom-in plots reveals that the high-order PVD-Net achieves superior accuracy within the internal layer region.
MethodGlobal ErrorsInner ErrorsTraining
 L2L^{2}L∞L^{\infty}L2L^{2}L∞L^{\infty}Time (s)
BL-PINNs [ 3 ]1.25×1001.25\times 10^{0}3.00×1003.00\times 10^{0}1.24×1001.24\times 10^{0}3.00×1003.00\times 10^{0}644
MSM-NN [ 51 ]9.42×10−19.42\times 10^{-1}1.97×1001.97\times 10^{0}9.48×10−19.48\times 10^{-1}1.97×1001.97\times 10^{0}1,091
Leading-order PVD-Net3.64×10−13.64\times 10^{-1}2.17×1002.17\times 10^{0}3.66×10−13.66\times 10^{-1}2.17×1002.17\times 10^{0}984
High-order PVD-Net4.03×10−2\mathbf{4.03\times 10^{-2}}2.82×10−1\mathbf{2.82\times 10^{-1}}4.06×10−2\mathbf{4.06\times 10^{-2}}2.82×10−1\mathbf{2.82\times 10^{-1}}1,995
Table 5: Comparison of prediction errors for PVD-Net on the internal layer problem after training for 100,000 epochs. Bold indicates the lowest error.
Figure 8: PVD-ONet for the internal layer problem. (Left): Predicted results of Leading-order PVD-ONet. (Right): Predicted results of High-order PVD-ONet. Our object is to learn the operator G:(a,b)↦uG:(a,b)\mapsto u .
MethodGlobal ErrorsInner Region ErrorsTraining
 Relative L2L^{2}L∞L^{\infty}Relative L2L^{2}L∞L^{\infty}Time (s)
PI-DeepONet [ 47 ]9.93×10−19.93\times 10^{-1}1.32×1001.32\times 10^{0}9.95×10−19.95\times 10^{-1}1.32×1001.32\times 10^{0}4,171
Leading-order PVD-ONet4.32×10−14.32\times 10^{-1}2.01×1002.01\times 10^{0}4.35×10−14.35\times 10^{-1}2.01×1002.01\times 10^{0}13,872
High-order PVD-ONet2.05×10−1\mathbf{2.05\times 10^{-1}}1.10×100\mathbf{1.10\times 10^{0}}2.07×10−1\mathbf{2.07\times 10^{-1}}1.10×100\mathbf{1.10\times 10^{0}}47,031
Table 6: Comparison of mean prediction errors and training time (100,000 epochs) for PVD-ONet on the internal layer problem. Bold indicates the lowest error.
Figure 9: Exact solution of the 2D coupled boundary layer problem and localized zoomed-in views of the left boundary layer, bottom boundary layer, and corner layer.
Figure 10: Global comparison of the exact solution, PINN, leading-order PVD-Net, and high-order PVD-Net.
Figure 11: Predicted solutions and absolute errors in the left boundary-layer region.
Figure 12: Predicted solutions and absolute errors in the bottom boundary-layer region.
Figure 13: Predicted solutions and absolute errors in the in the corner-layer region.
MethodGlobal Errorsxx -Layer Errorsyy -Layer ErrorsCorner ErrorsTraining
 Rel. L2L^{2}L∞L^{\infty}Rel. L2L^{2}L∞L^{\infty}Rel. L2L^{2}L∞L^{\infty}Rel. L2L^{2}L∞L^{\infty}Time (s)
PINN4.59×10−14.59\times 10^{-1}6.21×10−16.21\times 10^{-1}5.29×10−15.29\times 10^{-1}6.15×10−16.15\times 10^{-1}4.22×10−14.22\times 10^{-1}6.21×10−16.21\times 10^{-1}3.85×10−13.85\times 10^{-1}6.15×10−16.15\times 10^{-1}1,122
Leading-order PVD-Net1.04×10−21.04\times 10^{-2}1.33×10−11.33\times 10^{-1}3.27×10−33.27\times 10^{-3}1.33×10−11.33\times 10^{-1}5.59×10−35.59\times 10^{-3}1.33×10−11.33\times 10^{-1}1.44×10−21.44\times 10^{-2}1.33×10−11.33\times 10^{-1}2,864
High-order PVD-Net4.77×10−3\mathbf{4.77\times 10^{-3}}4.10×10−2\mathbf{4.10\times 10^{-2}}2.51×10−3\mathbf{2.51\times 10^{-3}}2.78×10−2\mathbf{2.78\times 10^{-2}}2.80×10−3\mathbf{2.80\times 10^{-3}}2.78×10−2\mathbf{2.78\times 10^{-2}}6.46×10−3\mathbf{6.46\times 10^{-3}}4.10×10−2\mathbf{4.10\times 10^{-2}}11,792
Table 7: Comparison of prediction errors for the two-dimensional coupled boundary-layer problem after training for 100,000 epochs. The bolded values indicate the lowest errors among all methods.
Figure 14: Comparison of full-domain solutions and corner layer regions for two test samples using PI-DeepONet, leading-order PVD-ONet, and high-order PVD-ONet.
MethodGlobal Errorsxx -Layer Errorsyy -Layer ErrorsCorner ErrorsTraining
 Rel. L2L^{2}L∞L^{\infty}Rel. L2L^{2}L∞L^{\infty}Rel. L2L^{2}L∞L^{\infty}Rel. L2L^{2}L∞L^{\infty}Time (s)
PI-DeepONet [ 47 ]2.79×10−12.79\times 10^{-1}6.05×10−16.05\times 10^{-1}3.30×10−13.30\times 10^{-1}5.76×10−15.76\times 10^{-1}2.22×10−12.22\times 10^{-1}6.05×10−16.05\times 10^{-1}2.34×10−12.34\times 10^{-1}5.76×10−15.76\times 10^{-1}1,070
Leading-order PVD-ONet9.30×10−39.30\times 10^{-3}6.29×10−26.29\times 10^{-2}1.33×10−21.33\times 10^{-2}5.22×10−25.22\times 10^{-2}1.22×10−21.22\times 10^{-2}6.29×10−26.29\times 10^{-2}5.69×10−35.69\times 10^{-3}3.79×10−23.79\times 10^{-2}13,280
High-order PVD-ONet3.62×10−3\mathbf{3.62\times 10^{-3}}2.77×10−2\mathbf{2.77\times 10^{-2}}4.84×10−3\mathbf{4.84\times 10^{-3}}2.77×10−2\mathbf{2.77\times 10^{-2}}4.68×10−3\mathbf{4.68\times 10^{-3}}2.28×10−2\mathbf{2.28\times 10^{-2}}1.64×10−3\mathbf{1.64\times 10^{-3}}2.11×10−2\mathbf{2.11\times 10^{-2}}25,296
Table 8: Comparison of different methods for the two-dimensional coupled problem after 10,000 training epochs. Bold indicates the lowest error.
ExampleLearned λ\lambda
Second-order equation with constant coefficients1.0100
Second-order equation with variable coefficients1.0014
Internal layer problem1.0131
Table 9: Learned scaling exponents for different examples using 10 data points. The exact value is λ=1\lambda=1 .
Figure 15: Log–log plots of the L2L^{2} and L∞L^{\infty} errors versus ε\varepsilon . (Left): PINN shows no clear convergence. (Middle and right): The L2L^{2} and L∞L^{\infty} errors of PVD-Net exhibit asymptotic convergence, with higher rates achieved by the high-order formulation.
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