PVD-ONet: Deep Operator Network for Boundary Layers

• PVD-ONet is a deep operator learning framework that integrates classical matched asymptotic expansions to solve singularly perturbed boundary layer problems.
• It decomposes solutions into inner (boundary layer) and outer (bulk) expansions, using neural networks to enforce precise matching conditions.
• By generalizing finite-parameter PVD-Net to an operator-learning setting, it enables rapid, data-free evaluations across families of boundary value problems even in extreme regimes.

The Prandtl–Van Dyke Deep Operator Network (PVD-ONet) is a neural-operator-based framework designed to solve singularly perturbed boundary layer problems, embedding matched-asymptotic analysis into deep operator learning. PVD-ONet leverages both Prandtl’s leading-order and Van Dyke’s high-order matching principles to decompose solutions into inner (boundary layer) and outer (bulk) expansions, enforcing rigorous matching at their interface. This approach enables stable and highly accurate predictions, even in the challenging $\varepsilon \to 0$ limit, a regime notorious for causing convergence failures in standard physics-informed neural networks (PINNs) and physics-informed DeepONet (PI-DeepONet) architectures. PVD-ONet generalizes the PVD-Net finite-parameter architecture to an operator-learning setting, enabling fast, data-free evaluation of solution operators for families of boundary value problems without retraining (Sun et al., 29 Jul 2025).

1. Mathematical Framework of Singularly Perturbed Boundary Layers

PVD-ONet is constructed for problems such as the one-dimensional convection–diffusion boundary-layer equation: $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$ supplemented with Dirichlet boundary conditions: $u(0) = \alpha, \quad u(1) = \beta,$ where $\varepsilon \ll 1$ denotes the singular perturbation that induces steep gradients ("layers") near domain boundaries. The method extends conceptually to higher dimensions, for example,

$$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$

but primary exposition and numerical tests focus on the prototypical 1D ODE (Sun et al., 29 Jul 2025).

2. Matched Asymptotic Expansions: Prandtl and Van Dyke Principles

Classical matched asymptotic expansions underpin the architecture:

2.1 Outer Expansion: Away from boundary layers, a regular expansion

$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$

yields, at leading and first order,

$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$

Boundary condition: $\varphi_0(1)=\beta$; higher-order terms vanish at $x=1$.

2.2 Inner Expansion: Near the layer (typically at $x=0$), the problem is rescaled via the stretched variable $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$0, seeking

$$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$1

The leading equation in the layer is

$$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$2

2.3 Prandtl’s Matching: Enforces leading-order consistency at the interface: $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$3

2.4 Van Dyke’s Matching: Systematically matches higher-order terms, ensuring all terms in the asymptotic series agree in the common region.

2.5 Composite Solution: Prandtl’s composite combines expansions: $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$4 with analogous constructions for higher-order (Van Dyke) corrections.

3. PVD-Net Neural Network Architectures

PVD-Net translates these expansions into composite physics-informed neural networks, available in two principal variants:

3.1 Leading-order PVD-Net: Comprises two subnetworks:

• Outer net $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$5, approximating $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$6;
• Inner net $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$7, approximating $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$8.

A matching penalty at $$\varepsilon\,\frac{d^2u}{dx^2}(x) + a(x)\,\frac{du}{dx}(x) + b(x)\,u(x) = 0, \quad x\in(0,1),$$9, $u(0) = \alpha, \quad u(1) = \beta,$0, is enforced in the loss function. The composite prediction is assembled as

$u(0) = \alpha, \quad u(1) = \beta,$1

3.2 High-order (Van Dyke) PVD-Net: Five subnetworks encode second-order asymptotics: $u(0) = \alpha, \quad u(1) = \beta,$2 Van Dyke matching terms are included in the loss, with the composite solution subtracting the matching polynomial.

4. PVD-ONet: Neural Operator Construction

PVD-ONet generalizes PVD-Net to operator learning using DeepONet modules in place of fully connected networks. Each DeepONet realizes a mapping

$u(0) = \alpha, \quad u(1) = \beta,$3

where the branch net encodes the problem data $u(0) = \alpha, \quad u(1) = \beta,$4, and the trunk net maps the location ($u(0) = \alpha, \quad u(1) = \beta,$5 or $u(0) = \alpha, \quad u(1) = \beta,$6).

4.1 Leading-order PVD-ONet: Uses outer and inner DeepONets $u(0) = \alpha, \quad u(1) = \beta,$7 and $u(0) = \alpha, \quad u(1) = \beta,$8, with training loss terms that parallel the leading-order PVD-Net but are summed over $u(0) = \alpha, \quad u(1) = \beta,$9 sampled problem data $\varepsilon \ll 1$0.

4.2 High-order PVD-ONet: Employs five DeepONet modules ($\varepsilon \ll 1$1), constructing composite solution operators with higher-order matching.

Inference involves a single forward pass for new boundary data $\varepsilon \ll 1$2, evaluating $\varepsilon \ll 1$3 and $\varepsilon \ll 1$4, then composing with the matching formula—no retraining is required for out-of-sample problems in the boundary data family.

5. Training Protocols and Implementation

• Collocation: $\varepsilon \ll 1$5 (outer domain) and $\varepsilon \ll 1$6 (inner stretched layer, $\varepsilon \ll 1$7); matching at $\varepsilon \ll 1$8.
• Network Dimensions: Leading-order: two networks, 5 layers $\varepsilon \ll 1$9 100 neurons (1000 neurons total); High-order: five networks, 5 layers $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$0 40 neurons (also 1000 total).
• Activation: SiLU functions throughout.
• Optimizer: Adam, learning rate $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$1, 100,000 iterations, best validation checkpoint retained.
• Normalization: Scale all coordinates and boundary data to $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$2.
• Matching Parameter: $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$3 approximates $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$4 for practical training.
• Automatic Differentiation: Computes all residuals, including matching and equation residuals.

6. Comparative Accuracy and Robustness

Performance on constant-coefficient test problems ($$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$5, $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$6, $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$7, $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$8) demonstrates marked superiority of PVD architectures over both traditional BL-PINN, MSM-NN, and PI-DeepONet, especially as $$\varepsilon\,\Delta u(x) + \mathbf{b}(x)\cdot\nabla u(x) + c(x)\,u(x) = f(x), \quad u|_{\partial\Omega} = g,$$9 decreases:

Method	Global $$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$0 Error	Global $$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$1 Error
BL-PINN	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$2	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$3
MSM-NN	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$4	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$5
Leading-order PVD-Net	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$6	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$7
High-order PVD-Net	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$8	$$u^o(x;\varepsilon) = \sum_{k=0}^\infty \varepsilon^k\,\varphi_k(x)$$9
PI-DeepONet	$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$0	$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$1
Leading-order PVD-ONet	$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$2	$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$3
High-order PVD-ONet	$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$4	$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$5

Standard PINN/DeepONet degrade in sharp layer regimes; PVD frameworks maintain stability and accuracy across all error metrics, including global $$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$6, $$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$7, inner-region, and interface (junction) errors (Sun et al., 29 Jul 2025).

7. Practical Guidelines and Theoretical Implications

• Selection of Model Order: Leading-order (stability-focused, 2 networks/DeepONets) offers rapid training for moderate accuracy; high-order (Van Dyke, 5 networks/DeepONets) delivers reduced error (5–10$$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$8 lower), essential for resolving fine-scale layers or gradient-sensitive observables.
• Matching Parameter $$\mathcal{O}(\varepsilon^0):\quad a(x)\,\varphi_0'(x)+b(x)\,\varphi_0(x)=0.$$9: Should approximate $\varphi_0(1)=\beta$0, but excessive size may hinder convergence during training.
• Architecture Balancing: Total neuron count is fixed across leading/high-order designs to allow for controlled comparison of convergence and accuracy.
• Operator Advantage: PVD-ONet enables rapid generalization—one training suffices for a family of boundary data—at the cost of greater training time and data volume relative to fixed-instance methods.
• Domain of Applicability: The method remains robust as $\varphi_0(1)=\beta$1, where traditional PINN and DeepONet architectures are prone to numerical instability or convergence failure.

PVD-ONet exemplifies the fusion of classical asymptotic boundary layer analysis with contemporary operator learning, providing a principled, architecture-level approach for multi-scale elliptic and convection–diffusion equations, especially those characterized by boundary-layer separation and sharp interior gradients (Sun et al., 29 Jul 2025).