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

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. PVD-ONet generalizes PVD-Net to the operator learning setting by assembling multiple DeepONet modules, directly mapping initial conditions to solution operators and enabling instant predictions for an entire family of boundary layer problems without retraining. Numerical experiments (second-order equations with constant and variable coefficients, and internal layer problems) show that the proposed methods consistently outperform existing baselines. Moreover, beyond forward prediction, the proposed framework can be extended to inverse problems. It enables the inference of the scaling exponent governing boundary layer thickness from sparse data, providing potential for practical applications.

Figures (14)

• 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 $\Omega_i$ and an outer region $\Omega_o$. By introducing the transformation $\xi = \frac{x - x_0}{\varepsilon}$, the boundary layer region $\Omega_i$ is mapped to an amplified region $\Omega_i^\prime$ in the stretched coordinate.
• Figure 2: Comparison of predicted results in the boundary layer region between our method and the baseline (BL-PINNs), using the representative boundary layer problem $\varepsilon u^{\prime\prime}+u^{\prime}+u=0$ as an example. (Left): Global prediction results of BL-PINNs. The blue solid line represents the analytical solution; the red dashed line indicates the predicted inner solution, and the green dashed line indicates the predicted outer solution. (Middle): Localized enlarged view of BL-PINNs in the boundary layer region, which shows a significant accuracy loss in the transition region. (Right): A localized zoomed-in view of our method (blue solid line: analytical solution; red dashed line: predicted solution). It can be seen that our method is able to realize smooth transitions and significantly improve the accuracy.
• Figure 3: The network architecture of PVD-Net. The PVD-Net consists of two distinct configurations: a leading-order version and a higher-order version. The leading-order PVD-Net employs two neural networks—--an outer network and an inner network—--to solve boundary layer problems through Prandtl's matching conditions, with the loss function computed accordingly. For the higher-order PVD-Net, the framework utilizes five neural networks in a hierarchical manner: two networks are dedicated to the Leading-order approximation, while three networks handle the first-order approximation. This novel architecture implements the Van Dyke matching principle to achieve asymptotic matching between solutions of different orders, with the loss function computed accordingly. Notably, within this architecture, one of the first-order approximation networks naturally reduces to the Leading-order case, maintaining consistency across approximation orders. Finally, both leading-order and high-order PVD-Net use composite solutions to obtain the output, thus guaranteeing smooth transitions in the solution.
• Figure 4: The architecture of Physics-informed DeepONet. The network consists of a branch network that encodes the input function $v$ sampled at fixed sensor locations $\{x_1,x_2,\dots,x_M\}$, and a trunk network that takes coordinate $\zeta$ as input. The outputs of the two networks are combined by dot product to approximate the solution operator. Automatic differentiation is used to compute PDE residuals and enforce physical constraints.
• Figure 5: The architecture of PVD-ONet. Similar to PVD-Net, PVD-ONet also consists of two approximation versions. PVD-ONet adopts DeepONet—a more expressive framework for operator learning. For the leading-order PVD-ONet, the model comprises two networks—--an inner and an outer network—--that learn a family of boundary layer problems. In contrast, the high-order PVD-ONet employs five networks to capture finer-scale structures.

Theorems & Definitions (5)

• Remark 1
• Remark 2
• Theorem 1
• Remark 3
• proof