Title: Hybrid Boundary Physics-Informed Neural Networks for Solving Navier-Stokes Equations with Complex Boundary
Venue: NeurIPS 2025
Accepted: yes
Keywords: Physics-informed neural networks

Navier-Stokes equations

complex boundary conditions
Abstract: Physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations (PDE), yet solving the Navier-Stokes equations (NSE) with complex boundary conditions remains a challenging task. In this paper, we introduce a novel Hybrid Boundary PINN (HB-PINN) method that combines a pretrained network for efficient initialization with a boundary-constrained mechanism. The HB-PINN method features a primary network focused on inner domain points and a distance metric network that enhances predictions at the boundaries, ensuring accurate solutions for both boundary and interior regions. Comprehensive experiments have been conducted on the NSE under complex boundary conditions, including the 2D cylinder wake flow and the 2D blocked cavity flow with a segmented inlet. The proposed method achieves state-of-the-art (SOTA) performance on these benchmark scenarios, demonstrating significantly improved accuracy over existing PINN-based approaches.

Hybrid Boundary Physics-Informed Neural Networks for Solving Navier-Stokes Equations with Complex Boundary

Abstract

Physics-informed neural networks (PINN) have achieved notable success in solving partial differential equations (PDE), yet solving the Navier-Stokes equations (NSE) with complex boundary conditions remains a challenging task. In this paper, we introduce a novel Hybrid Boundary PINN (HB-PINN) method that combines a pretrained network for efficient initialization with a boundary-constrained mechanism. The HB-PINN method features a primary network focused on inner domain points and a distance metric network that enhances predictions at the boundaries, ensuring accurate solutions for both boundary and interior regions. Comprehensive experiments have been conducted on the NSE under complex boundary conditions, including the 2D cylinder wake flow and the 2D blocked cavity flow with a segmented inlet. The proposed method achieves state-of-the-art (SOTA) performance on these benchmark scenarios, demonstrating significantly improved accuracy over existing PINN-based approaches.

1 Introduction

Fluid mechanics is an important field in science and engineering that deals with the study of the motion of liquids and gases. The NSE are basic partial differential equations that describe the dynamic behavior of viscous fluids, which are highly nonlinear partial differential equations and are widely used in aerodynamics, meteorology, oceanography, and industrial process simulations Munson et al. [1]. Although CFD methods have achieved considerable maturity, challenges such as mesh generation persist and can lead to numerical instability and reduced accuracy[2]. In recent years, PINNs, which integrate physical prior knowledge with deep learning [3], have emerged as a notable class of surrogate models for computational fluid dynamics.

PINN represent a deep learning methodology that incorporates physical constraints by embedding the governing physical equations and boundary conditions into the neural network's loss function. First proposed by Raissi et al. in 2019 [4], PINN eliminate the need for mesh generation, a requirement

inherent to traditional CFD solvers, while enabling more efficient resolution of inverse problems through data integration.

However, for fluid models with complex boundary conditions, conventional PINN methods often struggle to accurately approximate both the boundary conditions and the partial differential equations [5]. For conventional PINN, boundary conditions and initial conditions are explicitly embedded into the loss function, which is trained simultaneously with the PDE loss [4, 6]. A notable limitation of this loss formulation lies in its frequent failure to ensure simultaneous minimization of the PDE residual loss and the boundary condition loss. This issue becomes particularly pronounced when boundary conditions exhibit high complexity, thereby compromising the accuracy of the resultant solutions [7, 8, 9].

Here, to address the difficulty in balancing boundary condition losses and equation losses in complex boundary flow field problems, we propose a composite network approach specifically for complex boundary issues, named HB-PINN. By independently constructing a solution network that strictly enforces boundary conditions and a boundary-oriented distance function network, this method ensures applicability to diverse fluid dynamics problems with geometrically complex boundaries. To demonstrate the effectiveness of the proposed method, we conducted tests on two-dimensional incompressible flow fields under different boundary conditions, including steady-state and transient cases.

The main contributions of this work are as follows:

We propose a new PINN approach named HB-PINN that are embedded the boundary conditions to solve flow fields in regions with irregular obstructing structures.

A power function is proposed to refine the distance metric for handling complex boundary conditions, thereby improving prediction accuracy.

Extensive experimental validations on NSE with complex boundary conditions-such as the 2D cylinder wake problem and the 2D obstructed cavity flow with a segmented inlet-demonstrate the superior accuracy of the proposed model compared to existing PINN methods, thereby establishing new benchmarks.

2 Related Works

PINN methods for PDE. PINN were initially proposed by Raissi et al.[4]. The methods have been extended to diverse domains, including: fluid mechanics[10, 11], medical applications[12, 13], heat transfer[14, 15], and materials science[16, 17]. Furthermore, multiple PINN variants have been developed based on PINN, such as: Modified Fourier Network PINN (MFN-PINN)[18], hpVPINN[19], Conservative PINN (CPINN)[20]. These variants enhance the generalizability and accuracy of PINN in solving heterogeneous PDE systems.

Loss conflict problems. The PINN framework explicitly embeds boundary and initial conditions into the loss function, and the resultant loss conflict problem has garnered significant attention from the research community. A pivotal development by Lu et al. (2021) [21] proposes a distance-informed ansatz that provably satisfies boundary conditions through analytically constructed solutions. This architecture enables exclusive optimization of PDE residuals during training. SA-PINN [22] proposes a spatially adaptive weight matrix to dynamically mitigate gradient conflicts between region-specific loss components during optimization. XPINN [23], as a generalized extension of PINN, introduces an interface-informed decomposition framework for solving nonlinear PDE on arbitrarily complex geometries. Although these methods show significant improvements in loss balancing compared to baseline PINN models, they still suffer from inaccuracies when handling problems with highly complex boundary conditions.

Complex boundary problems. Botarelli et al. [24] employed a Modified Fourier Network (MFNPINN) to solve flow field problems with complex boundaries, demonstrating its superiority through comparisons with Multi-Layer Perceptron-based PINN (MLP-PINN). Sukumar et al. [25] proposed an R-function-based distance function construction method for enforcing boundary conditions in PINN, which significantly improves convenience in first-order derivative-dominated problems. However, the complex boundary analytical distance functions (ADF) constructed by R-functions are not natural functions.

To address the loss term conflicts induced by complex boundary conditions, this work decouples the boundary constraints into two sub-functions: a particular solution function that strictly enforces boundary conditions while being weakly trained on the governing equations, and a distance function describing the spatial proximity to boundaries. Furthermore, to ensure global differentiability of the distance function, we formulate it via a DNN architecture.

3 Methodology

3.1 Incompressible Navier-Stokes Equations

The incompressible Navier-Stokes equations, excluding external body forces, serve as the governing equations for all flow cases in this study, expressed as:

In the equations, u , p , ρ , and ν denote the velocity vector, fluid pressure, density and dynamic viscosity coefficients, respectively. For incompressible flows, ρ and ν are characteristic fluid parameters that remain spatially and temporally invariant.

3.2 HB-PINN Approach

The architecture of our proposed method is illustrated in Fig.1. The framework consists of three subnetworks: N P , N D , and N H . The N P subnetwork (Particular Solution Network) is trained to satisfy boundary conditions, while the N D subnetwork (Distance Metric Network) learns boundarycondition-aware weights by encoding the distance from interior points to domain boundaries. The N H subnetwork (Primary Network) is dedicated to resolving the governing PDE. During the final training phase, only the parameters of N H are optimized, whereas N P and N D remain fixed as pre-trained components. This design effectively addresses challenges involving hybrid interior-exterior boundary configurations. Therefore, we term this methodology Hybrid Boundary Physics-Informed Neural Networks.

In our model, the constraints on the variables are formulated as follows:

here, q ( · ) denote the physical quantities of interest (e.g., u, v , and p in a 2D flow model); P q represents an additional solution function that satisfies the boundary conditions; D q is the distance function; H q is the output of the primary network. The distance function D q takes a value of 0 at the domain boundaries and rapidly increases to 1 as it moves away from the boundaries. Functionally, D q acts as a weight for H q , modulating its influence across the computational domain.

Under this formulation, the distance function D q enforces strict boundary condition compliance by evaluating to zero at domain boundaries while smoothly transitioning to H q , which satisfies the governing equations in interior regions. When boundary conditions and geometric configurations are simple, both P q and D q can be analytically expressed. However, analytical expressions for P q and D q are generally infeasible for complex geometries. Consequently, we leverage three dedicated deep neural networks (DNN) , the previously defined N P , N D , and N H , to parameterize these components. The composite solution for 2D incompressible flows is thus constructed as:

Sub-network N P

The subnetwork N P adopts the same architecture as conventional PINN, with outputs comprising the velocity components u, v , and pressure p . The loss function for training N P is formulated by integrating residuals of the NSE, boundary conditions (BC), and initial conditions (IC), expressed as:

where and

Here, λ i ( i = 1 , 2 , 3) denote the loss weighting coefficients corresponding to each term. Adjusting these weights allows biasing the network's training dynamics to enhance solution accuracy. Our N P subnetwork prioritizes boundary and initial condition enforcement by assigning λ 2 , λ 3 ≫ λ 1 (e.g., λ 1 = 1 ; λ 2 , λ 3 = 1000 ), enabling preliminary PDE training while strictly satisfying boundary constraints.

By decoupling the training objectives, the subnetwork N P focuses exclusively on enforcing boundary conditions through loss weight modulation, without imposing strict constraints on governing equation residuals. This targeted approach circumvents the multi-loss balancing challenges inherent to conventional PINN, while delegating full PDE resolution to the primary network N H .

Sub-network N D

The N D subnetwork is a shallow DNN designed to construct distance functions. For each quantity q of interest, we introduce a distance metric network N D q . To train this network, sampling points ( x, t ) are firstly collected from the space-time domain Ω × [0 , T ] . The network is then supervised to approximate the signed distance function ˆ D q ( x, t ) between these points and the boundary associated with q , formulated as:

PretrainedNp

P

Pp

=Pu + Du·u

=P+D·

p=Pp+Dp·p

0H/00H

u = (u,0)T

CH =V·ul|²+(ou/ot) + (u·V)u+(1/p)Vp-V²u ²

Pretrained ND

D

Particular solution network Np

Distance metric networkNp

0Cp,/00g

CpDe = [V u]² +[(ou/0r) + (u- V)u + (1/)VP,  vV²u]

u= (P,P,)T

Np.(z.y:0p.)

Cpc = [u uscl2

Cnc = [u ucll?

D = 1  (1  d/max(d,)。

E{α,r,P

Cp = AμCpoe + ACtc + AsCBc

0Cp,/06p,

ND,(x,y;0D,)

00/a7

0Cp/00p

In order to ensure the network outputs approach 0 near domain boundaries and rapidly increase to 1 in regions far from boundaries, while maintaining dominance of the primary network N H q 's predictions in interior regions, we employ a power-law function f ( ˆ D q ) derived from the distance metric ˆ D q as the training labels for the subnetwork N D q . The power-law function f ( ˆ D q ) is defined as follows:

The parameter α is a positive value controlling the growth rate of the function. The curves exhibit steeper gradients near boundaries and smoother transitions in regions distant from boundaries. Larger α values enhance the sensitivity of the distance metric network to boundary-proximal points while diminishing its response to interior points, thereby reducing computational demands. However, excessively large α values may lead to erroneous predictions of near-boundary points as having distance values close to 1, which undermines the influence of the specialized solution network in critical regions. To accommodate varying boundary complexities across models, α can be tuned to maintain the steepness of the power-law function within a reasonable range. The loss function for the distance metric network is defined as:

Here, D q represents the network's output, supervised by the training labels f ( ˆ D q ) . This approach ensures robust training of N D q even under highly complex boundary conditions, while simultaneously enhancing the accuracy of the primary network's training outcomes.

Sub-network N H

The N H subnetwork also employs a DNN architecture but with a larger scale compared to the N P and N D subnetworks. Through pre-training of the N P and N D subnetworks, boundary conditions are strictly enforced, allowing N H to focus solely on minimizing the governing equation residuals. For the 2D incompressible NSE, the loss function of the N H subnetwork is defined as:

While this formulation shares the same structure as Eq.7, the variables ˆ u and ˆ p are derived from the modified outputs of Eq.2. This training strategy enables the N H subnetwork to focus exclusively on PDE compliance during optimization, effectively eliminating the conflicting gradients between equation residuals and boundary condition losses that commonly plague conventional PINN.

4 Experiments

To evaluate the effectiveness of the proposed approach, three two-dimensional incompressible flow cases are investigated, including two steady-state cases and one transient case, to demonstrate the general applicability of our method across different flow fields.

The steady-state and transient incompressible Navier-Stokes equations were solved via the finite element method (FEM), with mesh density ensuring solution accuracy and numerical stability. A time step size of ∆ t = 0 . 01 was adopted for transient simulations. The CFD results obtained under these configurations were utilized as ground truth (GT), the proposed HB-PINN is benchmarked against the conventional soft-constrained PINN (sPINN) [4], hard-constrained PINN (hPINN) [21], modified Fourier Network PINN (MFN-PINN) [18, 24], extended PINN (XPINN) [23], self-adaptive PINN (SA-PINN) [22], and PirateNets [26]. Notably, the hPINN, proposed by Lu et al.[21], exhibits erratic behavior in scenarios with complex boundary conditions, often failing to produce reliable results for comparative analysis. To ensure comparability of results, the comparative analysis of the hPINN method in this study selectively relaxes the enforcement of boundary conditions in specific subdomains, as exemplified by the near-inlet wall regions in Case 2 ( 0 < x < 0 . 4 , y = 0 & y = 1 ). The detailed rationale for relaxing boundary conditions is provided in Appendix D. Additionally, to

assess the impact of our HB-PINN, we conducted three ablation studies on Case 2, the results are provided in Appendix C.

In this section, the experimental setup is introduced first, followed by three case studies: steady-state two-dimensional flow around a cylinder, steady-state flow in a segmented inlet with an obstructed square cavity, and transient flow in a segmented inlet with an obstructed square cavity.

4.1 Experimental Settings

As illustrated in Fig. 1, the proposed HB-PINN takes space-time coordinates as inputs and outputs velocity components ( u , v ) and pressure p . In the implementation phase, the training process employs a distributed architecture: each subnetwork ( N P , N D , and N H ) utilizes three independent DNN dedicated to variables u , v , and p , resulting in a total of nine distinct DNN. After obtaining separate predictions for u , v , and p , these outputs are integrated through Eq. 3 to reconstruct the composite solution. For detailed training protocols refer to Appendix B.

In the models discussed in this study, the NSE are non-dimensionalized using characteristic scales. For the three benchmark cases analyzed in the main text, the Reynolds number is uniformly set to Re = 100 as the baseline configuration. Comparative studies at higher Reynolds numbers ( Re = 500 , 1000 , 2000 ) are systematically documented in Appendix E.

"dN

NDu

1.00

0.75

ND

0.50

p

0.25

0.00

SPINN

hPINN

MFN-PINN

2.0

X-PINN

SA-PINN

PirateNet

1.5

1.0

HB-PINN(ours)

CFD(GT)

0.5

0.0

4.2 Case 1: Two-dimensional flow around a cylinder

The flow past a circular cylinder represents a canonical benchmark in fluid mechanics(see Appendix A.1 for detailed geometric specifications). The boundary conditions are specified as follows:

Following Eq.4, the loss function for the N P subnetwork in the proposed model is formulated as:

Here, Ω denotes the interior of the geometric region, while ∂ Ω denotes the boundary. The loss weight λ 1 for the equation part is set to 1, and the loss weight λ 2 for the boundary part is set to 1000, with the aim of making the network focus more on the training of boundary conditions.

The outputs of the N P and N D networks are shown in Fig.2. The final modification to the variable result is as follows:

Since both v and p are set to 0 at the boundary, there is no need to configure the boundary results separately; they are simply constrained by the distance function.

In the case of 2D cylindrical pipe flow model, the qualitative results from sPINN, hPINN, MFNPINN, XPINN, SA-PINN, PirateNet, HB-PINN are shown in Fig.3. The residuals of these methods compared with CFD results are shown in Fig.4. More detailed error metrics are summarized in the 'Case1' section of Table 1.

SPINN

hPINN

MFN-PINN

1.25

1.00

X-PINN

SA-PINN

PirateNet

0.75

0.50

HB-PINN(ours)

0.25

0.00

4.3 Case 2: steady-state flow in a segmented inlet with an obstructed square cavity

The model incorporates two rectangular obstructions positioned along the top and bottom walls of a square cavity, with a segmented inlet configuration (see Appendix A.2 for detailed geometric specifications). The boundary conditions are configured as follows:

The output results of the boundary condition solution function network related to u and the distance function network for u , v , and p are shown in Fig.5.

"dN

NDu

MaN

ND

1.00

0.75

0.50

0.25

0.00

The velocity distributions for case 2 from sPINN, hPINN, MFN-PINN, SA-PINN, XPINN, PirateNet and our HB-PINN are shown in Fig.6. Compared to existing methods, the HB-PINN framework demonstrates superior capability in simulating models with complex inflow boundary conditions, achieving a MSE reduction of an order of magnitude relative to CFD benchmarks. The residuals of these methods compared with CFD results are shown in Fig. 7. More detailed error metrics are summarized in the 'Case2' section of Table 1.

Based on Case 2, we conduct ablation studies on two critical pre-trained sub-networks by evaluating the outcomes of separately employing the specific solution network and the distance metric network. Here, 'mP' denotes using only the particular solution network, while 'mD' refers to using only the distance metric network.

Under the mP approach, the distance function is normalized solely based on the distance to the boundary without adjustment using power functions, resulting in lower accuracy within the domain. Under the mD approach, while internal errors are reduced during training, the enforcement of

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

0.75

HB-PINN(ours)

0.50

XPINN

SA-PINN

PirateNet

0.25

0.00

boundary conditions is imprecise, which affects the final results. These tests demonstrate that both constructive functions in our method contribute positively to the solution process, further validating the necessity of applying both functions simultaneously. The comparison of residuals is shown in Fig.8. For a more detailed error analysis, please refer to Table2 in Appendix C.

mP

mD

hPINN

HB-PINN(ours)

0.8

0.8

0.8

0.8

2.0

0.6

0.6

0.6

0.6

1.5

0.4

0.4

0.4

0.4

CFD(GT)

1.0

0.2

0.2

0.2

0.2

0.8

0.5

0.25

50.500.75

0.25 0.50 0.75

0.25 0.50 0.75

0.25 0.50 0.75

0.6

0.0

mP Error

hPINN Error

HB-PINN Error

0.4

0.75

0.8

0.8

0.8

0.8

0.2

0.50

0.6

0.6

0.6

0.6

0.25 0.50 0.75

0.4

0.4

0.4

0.4

0.25

0.2

0.2

0.2

0.2

0.00

0.25 0.50 0.75

0.25 0.50 0.75

0.25 0.50 0.75

0.25 0.50 0.75

4.4 Case 3: transient flow in a segmented inlet with an obstructed square cavity

The difference between this model and case 2 lies in the consideration of the transient situation, which further increases the complexity of the boundary conditions. The boundary conditions are set

as follows: the inlet normal velocity u is 0.5, the outlet static pressure is 0, and the velocity at other boundaries is set to 0, satisfying the no-slip condition. The initial conditions are set with the velocity components u , v and the pressure p all equal to 0. Relative to Case 2, the addition of initial conditions:

The network structure parameters chosen for the transient case are the same as those for case 2. Fig. 9 shows the velocity results obtained using different methods at t = 1 . In the transient case, due to the increased complexity of the boundary conditions, the accuracy of the other methods significantly decreases, while HB-PINN maintains a high level of accuracy. The velocity results and the error compared to the CFD method are shown in Fig. 10. These results demonstrate the versatility of HB-PINN in both steadystate and transient situations.

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

4.5 Result Discussion

The quantitative results of the error metrics between CFD calculation and the predictions from sPINN, hPINN, MFN-PINN, SA-PINN, XPINN, PirateNet and HB-PINN for the three cases are presented in Table 1. Combined with the qualitative results shown in Fig.3,4,6,7,9,10, it is shown that our HB-PINN can accurately solve complex boundary flow problems with interior barriers, while the traditional soft-constrained PINN (sPINN) method and hard-constrained PINN (hPINN) method incur significant errors in solving such problems. The training results of SA-PINN and XPINN show improvements in accuracy; however, significant errors persist in the cases with more complex boundary conditions.

SPINN

hPINN

MFN-PINN

1.25

HB-PINN(ours)

1.00

0.75

XPINN

SA-PINN

PirateNet

0.50

0.25

0.00

5 Limitations

The optimal values of critical parameters in the methodology, such as the α parameter in the distance metric network and the training iterations of the particular solution network, are currently determined empirically. Further research is required to systematically identify their optimal configurations.

6 Conclusion

PINN hold great potential as surrogate models for fluid dynamics; however, their application is often hindered by the complexity of the NSEs and intricate boundary conditions. This paper introduces a novel method, HB-PINN, which features a primary network focused on inner domain points and a distance metric network that enhances predictions at the boundaries, ensuring accurate solutions for both boundary and interior regions. This makes the approach applicable to be adaptable to more complex fluid dynamics models. Through tests on steady-state two-dimensional flow around a cylinder, steady-state segmented inlet with obstructed square cavity flow, and transient segmented inlet with obstructed square cavity flow, our method outperforms typical PINN approaches by a large margin, establishing itself as a benchmark for solving NSEs with complex boundaries. Additional analysis is provided in the appendices to further illustrate the effectiveness of our method.

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A Supplementary Geometric Details

A.1 Case1:Flow around a Cylinder

三

1

0

5

Case 1 corresponds to the canonical flow past a circular cylinder. As schematically depicted in Fig.11. Following non-dimensionalization, the computational domain is configured with a length of 5 and a width of 1. A circular obstruction with diameter D = 0 . 2 is centered at ( x, y ) = (0 . 5 , 0 . 5) . The boundary conditions are defined as follows:

Inlet (Left Boundary): Horizontal inflow velocity u = 1 , v = 0 .

Outlet (Right Boundary): Static pressure fixed at p = 0 .

Top/Bottom Boundaries: No-slip walls ( u = v = 0 ).

This configuration corresponds to a Reynolds number Re = 100 , with the dimensionless density ρ = 1 . The dynamic viscosity ν is determined by the Reynolds number formula Re = uD ν , where u denotes the characteristic velocity and D the characteristic length.

A.2 Case2:Steady-State Flow in a Segmented Inlet with an Obstructed Square Cavity

1

三

0

1

The model features two rectangular obstructions positioned at the top and bottom walls of a square cavity with an edge length of 1 unit. As illustrated in Fig.12, the inlet is divided into three segments, and the obstructions have a width of 0 . 2 units and a height of 0 . 3 units. The boundary conditions are defined as follows:

Inlet: Normal inflow velocity u = 0 . 5 , v = 0 .

Outlet: Static pressure set to p = 0 .

Other Boundaries: No-slip conditions ( u = v = 0 ) on all walls and obstruction surfaces.

In Case 3 of the main text, the geometric configuration remains identical to that of Case 2, with the sole distinction lying in the incorporation of transient flow dynamics.

B Training Details

The framework is developed on the PyTorch machine learning platform, and training iterations are accelerated using a high-performance GPU.

B.1 Case1

Subnetwork N P :

Network architecture: Three separate deep neural networks (DNNs) for variables u , v , and p , each structured as [2] + 6 × [64] + [1]

Training epochs: 10 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: 1 × 10 -3

Subnetwork N D :

Network architecture: Three separate DNNs for u , v , and p , each structured as [2]+4 × [20]+[1]

Training epochs: 300 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

α = 10

Subnetwork N H :

Network architecture: Three separate DNNs for u , v , and p , each structured as [2]+6 × [128]+[1]

Training epochs: 300 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

B.2 Case2

Subnetwork N P :

Network architecture: Three separate deep neural networks (DNNs) for variables u , v , and p , each structured as [2] + 6 × [64] + [1]

Training epochs: 10 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: 1 × 10 -3

Subnetwork N D :

Network architecture: Three separate DNNs for u , v , and p , each structured as [2]+4 × [64]+[1]

Training epochs: 300 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

α = 5

Subnetwork N H :

Network architecture: Three separate DNNs for u , v , and p , each structured as [2]+6 × [128]+[1]

Training epochs: 500 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

B.3 Case3

Subnetwork N P :

Network architecture: Three separate deep neural networks (DNNs) for variables u , v , and p , each structured as [3] + 6 × [64] + [1]

Training epochs: 10 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: 1 × 10 -3

Subnetwork N D :

Network architecture: Three separate DNNs for u , v , and p , each structured as [3]+4 × [64]+[1]

Training epochs: 300 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

α = 5

Subnetwork N H :

Network architecture: Three separate DNNs for u , v , and p , each structured as [3]+6 × [128]+[1]

Training epochs: 500 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

B.4 Other PINN methods

Network architecture: Three separate DNNs for u , v , and p , each structured as [2]+6 × [128]+[1]

Training epochs: 500 , 000

Activation function: tanh

Optimizer: Adam

Learning rate: Initial learning rate of 1 × 10 -3 with a learning rate annealing strategy

Points per epoch: 20 , 000 (random sampling from domain)

C Ablation studies

The three ablation studies are as follows:

Comparative analysis of mP (particular solution network only) and mD (distance metric network only) against hPINN and HB-PINN, where mP and mD represent experimental configurations using isolated components of HB-PINN.

Impact assessment of varying pre-training epochs for the particular solution network ( N P ) on the final accuracy of the primary network.

Sensitivity analysis of the power-law exponent α in the distance metric network ( N D ), which governs the transition steepness of the distance function, to evaluate its influence on solution accuracy.

C.1 mP and mD

In this section, we evaluate the outcomes of separately employing the particular solution network and the distance metric network, where mP refers to using only the particular solution network and mD refers to using only the distance metric network. The experimental results are compared with hPINN and HB-PINN, and velocity predictions are shown in Fig.13.

mP

mD

2.00

0.8

0.8

0.6

0.6

1.75

0.4

0.4

1.50

0.2

0.2

1.25

hPINN

0.2

0.4

0.6

HB-PINN(ours)

0.8

0.2

0.4

0.6

0.8

CFD(GT)

1.00

0.75

80

0.8

8°0

0.6

0.6

0.6

0.50

0.4

0.4

0.4

0.25

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0.00

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0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

mP

mD

0.75

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0.50

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

hPINN

HB-PINN(ours)

80

0.8

0.25

0.6

0.6

0.4

0.4

0.2

0.2

0.00

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

Under the mP approach, the distance function is normalized solely based on the distance to the boundary without adjustment using power functions, resulting in lower accuracy within the domain.

Under the mD approach, while internal errors are reduced during training, the enforcement of boundary conditions is imprecise, which affects the final results. These tests demonstrate that both constructive functions in our method contribute positively to the solution process, further validating the necessity of applying both functions simultaneously. The comparison of residuals is shown in Fig.14.More detailed error metrics are summarized in the 'mP and mD' section of Table ?? .

C.2 Particular solution network under different training epochs

In the HB-PINN method, the particular solution network ( N P ) is trained with a shallow deep neural network (DNN) using a soft constraint approach, aiming to provide a pre-trained solution that satisfies the boundary conditions, by tuning the weights associated with the loss terms, the network's output is rigorously constrained to strictly satisfy the boundary conditions. When the weight of the boundary condition loss term in the loss function is set to an extremely high value (e.g., 1000), excessive training iterations may lead to overfitting in local boundary regions, thereby compromising the global smoothness of the network's output. To mitigate this issue, a strategy with limited training iterations is generally adopted. In this section, we supplement our analysis by investigating the impact of different training iterations (10,000, 30,000, and 50,000) of N P on the final training results. The velocity results and their residuals relative to the CFD method are shown in Fig.15,16.

10000_epoch

30000_epoch

2.00

0.8

0.8

0.6

0.6

1.75

0.4

0.4

1.50

0.2

0.2

1.25

0.2

50000_epoch

0.4

0.6

8'0

0.2

CFD(GT)

0.4

0.6

0.8

1.00

0.75

0.8

0.8

0.6

0.6

0.50

0.4

0.4

0.25

0.2

0.2

0.00

0.2

0.4

0.6

8'0

0.2

0.4

0.6

0.8

10000_epoch

30000_epoch

50000_epoch

0.20

8°0

0.8 -

0.8-

0.15

90

0.6

9'0

0.10

0.4

0.4

0.4 -

0.05

0.2

0.2 -

0.00

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0.4

0.6

0.8

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

As evidenced by the error metrics in Figure 15 and the 'Epoch for N P ' section of Table 2, limiting the training iterations of the particular solution subnetwork ( N P ) leads to lower global error within the computational domain and more rapid convergence of the primary network ( N H ). However, this configuration results in reduced precision in boundary regions compared to cases where N P undergoes extended training.Currently, the number of training iterations for N P is determined empirically, and further research is needed to balance accuracy and convergence speed optimally.

C.3 Distance metric network under different α

This section evaluates the impact of four critical parameter values ( α = 3 , 5 , 10 , 15 ) in the distance metric network on the predictive performance of the HB-PINN method. Notably, α = 5 corresponds to the parameter configuration adopted in Case 2 of the main text. The velocity prediction results, as illustrated in Fig.17, demonstrate the sensitivity of model accuracy to variations in α .

α=3

α=5

2.00

1.75

CFD(GT)

1.50

1.25

1.00

α=10

0.75

0.50

0.25

0.00

α=3

α=5

α=10

0.25

0.20

0.15

0.10

0.05

0.00

The parameter α governs the steepness of the transition in the distance function near boundaries. Excessively small α values (e.g., α = 3 ) result in insufficient sensitivity to boundary constraints, while overly large α values (e.g., α = 10 , 15 ) lead to excessively thin boundary layers, thereby amplifying training challenges in boundary regions. The velocity prediction residuals compared to CFD results and the statistical metrics of the mean squared error (MSE) are shown in Fig.18 and the α for N D section of Table2. These results indicate that both undersized and oversized α values degrade prediction accuracy. However, the selection of α currently remains empirically determined, and identifying an optimal α value warrants further investigation.

C.4 Error Records

D Supplementary Details on Boundary Condition Relaxation in the hPINN Method

In the hPINN method, using only boundary conditions to constrain the solution network N P is effective when the boundary conditions are simple. However, for complex boundaries, this approach leads to distorted or discontinuous outputs within the solution domain, especially near junctions between different boundary types.

Fig.19 presents the results of the boundary particular solution network N P from the original hPINN, the hPINN with partially relaxed boundary constraints, and the HB-PINN. It can be clearly observed that under the original hPINN framework, the output of the N P network exhibits irregular solutions within the domain. Since it is trained solely on boundary conditions, its internal results are uncontrollable. This significantly increases the training difficulty of the main network, even hindering

convergence - as we observed in our hPINN experiments: strict constraints on certain boundary regions had to be relaxed for the main network loss to decrease effectively. The loss records in Table3 fully demonstrate this point. In Relaxed hPINN, we relaxed the boundary constraints in the regions 0 < x < 0 . 4 , y = 0 & y = 1 , which indeed effectively promoted the reduction of loss during main network training. However, this compromise violates the requirement for full boundary condition satisfaction, which is unacceptable for solving physical models. This precisely highlights the dilemma of traditional hard-constraint methods when handling complex boundary problems. Therefore, we introduced the N P with weak equation constraints as an alternative solution aimed at more robustly handling complex boundary constraints.

Original hPINN

Relaxed hPINN

HB-PINN

0.5

0.8

0.8

0.8

0.4

0.6

0.6

0.6

0.3

0.4

0.4

0.4

0.2

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E Supplementary details of the transient case (Case 3).

In the main text of Case 3, we present the velocity results at t = 1 for the transient segmented inlet with obstructed square cavity flow model. At t = 1, the flow can be considered to have developed to a state approaching steady-state results. Fig.20,21,22,23 display the velocity results at several time

steps during the development of the transient model, specifically at t = 0.1, t = 0.3, t = 0.5, and t = 0.7, to demonstrate that our HB-PINN method achieves high accuracy across all time steps when solving the transient model. The errors in velocity relative to the CFD method are shown in Fig.24,25,26,27

The velocity results at different time steps and their corresponding mean square errors relative to the CFD method are quantitatively summarized in Table4.

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

1.25

HB-PINN(ours)

1.00

0.75

XPINN

SA-PINN

PirateNet

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

1.25

HB-PINN(ours)

1.00

0.75

XPINN

SA-PINN

PirateNet

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

1.25

HB-PINN(ours)

1.00

0.75

XPINN

SA-PINN

PirateNet

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

1.25

HB-PINN(ours)

1.00

0.75

XPINN

SA-PINN

PirateNet

0.50

0.25

0.00

F Cross-sectional profiles and time-history results in high-gradient regions

This section compares the velocity and pressure predictions of sPINN, PirateNet, and HB-PINN in high-gradient regions. Fig.28 displays the predictive results along the cross-section at y = 0 . 5 in Case 2 compared with CFD results, while Fig.29 presents the temporal distribution of predictions at the center point (0 . 5 , 0 . 5) in Case 3 against CFD references. Both regions represent areas with substantial gradients within their respective domains. The results demonstrate that HB-PINN achieves optimal prediction accuracy for all physical quantities in both test cases.

x"conponent of velocity

y"component of velocity

Pressure

0.8

OFP(NT)

Pirateliet

I6-PIMN(ivrs)

0.35

0.30

0840

Pivateliet

16P1MN (iurs)

PIN

3.0

2.5

0.25F

5↓199|94

0.20

2.0

F0. 15

1

0.10

1.0

0.05

0.00

0.5

Pirateet

sPIM

HBP1N (ours)

0.2

x"component of velocity

ycomponent of velocity

Pressure

0F9(GT)

IN

Piratefiet.

I6PINN(oors)

0. 14R

TETIE

(I5)00

Firatefiet

I6-PNN(ours)

0. 12

A1100|0

0. 8

0. 6

8 0.08

0. 10

0.06

ISIE

0FD(6T)

Piratelet

16-HN(aurs)

sPIMN

1D

4

0.04

Q.

0.02-1

0.00

4

G Steady-state flow in a square cavity with obstructed segmented inlet at different Reynolds numbers

As an extension of the comparative study on three flow models under the Reynolds number Re = 100 scenario in the main text, this supplementary investigation systematically evaluates model

performance at elevated Reynolds numbers ( Re = 500 , 1000 , 2000 ) using the Case 2 configuration, while strictly maintaining the original geometric structure and boundary conditions.

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

0.75

HB-PINN(ours)

0.50

XPINN

SA-PINN

PirateNet

0.25

0.00

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

0.75

HB-PINN(ours)

0.50

XPINN

SA-PINN

PirateNet

0.25

0.00

SPINN

hPINN

MFN-PINN

XPINN

2.00

1.75

1.50

1.25

1.00

SA-PINN

PirateNet

HB-PINN(ours)

CFD(GT)

0.75

0.50

0.25

0.00

SPINN

hPINN

MFN-PINN

0.75

HB-PINN(ours)

0.50

XPINN

SA-PINN

PirateNet

0.25

0.00

Fig.30,32,34 present the velocity predictions of different methods at Re = 500 , 1000 , and 2000 , respectively, while their residuals relative to high-fidelity CFD results are visualized in Fig.31,33,35. As shown in the MSE variation trends across Reynolds numbers (Fig.36), all methods exhibit rising

prediction errors with increasing Re . However, the proposed HB-PINN consistently achieves the lowest MSE values at every Reynolds number.

Quantitative error metrics under multiple evaluation criteria are tabulated in Table.5. Notably, HB-PINN achieves an order-of-magnitude reduction in MSE compared to conventional methods at Re = 500 and 1000 . Even under the most challenging Re = 2000 condition, HB-PINN demonstrates a notable accuracy improvement.

Model

SPINN

0.14

hPINN

MFN_PINN

SAPINN

0.12

XPINN

PirateNet

HB-PINN

0.10

E

0.08

M

0.06

0.04

0.02

0.00

500

1000

2000

Reynolds Number

H The more complex obstructed cavity flow model

2

To validate the capability of our Hybrid Boundary Physics-Informed Neural Network (HB-PINN) in handling geometrically complex scenarios, we extend Case 2 from the main text to a more intricate configuration. As illustrated in Fig.37, the computational domain is expanded to a rectangular cavity of width 2 and height 1 . Two types of staggered obstructions are embedded within the cavity:

Type A Obstructions: Width = 0 . 2 , Height = 0 . 5

Type B Obstructions: Width = 0 . 2 , Height = 0 . 25

The segmented inlet configuration (see Appendix E.1 for geometric specifics) imposes a normal inflow velocity u = 0 . 5 , while the outlet, positioned at the upper-right corner of the cavity, enforces a static pressure p = 0 . All other boundaries adhere to no-slip conditions ( u = v = 0 ).

SPINN

error_sPINN

0.8

0.6

1.720

0.4

0.2

1.505

HB-PINN(ours)

error_HB-PINN

1.290

0.8

1.075

0.6

0.860

0.4

0.2

0.645

0.25

0.50

0.75

1.00

1.251.50

1.75

0.25

0.500.75

1.00

1.25

1.501.75

0.430

X

CFD(GT)

X

0.215

0.000

0.25

0.500.75

1.251.501.75

X

Fig.38 compares the velocity prediction results of conventional soft-constrained PINN (sPINN) and the proposed HB-PINN in the extended obstructed cavity flow model. As the geometric and physical

complexity of the model increases, traditional PINN methods struggle to resolve conflicts between competing loss terms (e.g., boundary condition residuals vs. governing equation residuals), leading to significant performance degradation in velocity field predictions. In contrast, HB-PINN maintains robust training convergence and achieves consistently accurate results. More detailed error metrics are summarized in the 'Complex Cavity Flow' section of Table 6.

I Heat equation

To demonstrate the applicability of the HB-PINN method to other PDE systems, we evaluate its performance on a two-dimensional transient heat conduction problem, governed by the following equation:

T ( x, y, t ) represents the temperature field, where the thermal diffusivity α = k ρc is defined by the thermal conductivity k , density ρ , and specific heat capacity c . The geometric configuration, shown in Fig.39, features a square domain with edge length 1 , containing four square heat sources (edge length 0 . 1 ) centered at coordinates (0 . 3 , 0 . 3) , (0 . 3 , 0 . 7) , (0 . 7 , 0 . 3) , and (0 . 7 , 0 . 7) . These heat sources are maintained at a dimensionless temperature of 1 , while all external boundaries are fixed at 0 .

1

口

口

口

口

0

1

J Lid-driven Cavity flow

In this section, we evaluate the performance of HB-PINN in lid-driven cavity (LDC) flow, a classical benchmark problem in computational fluid dynamics. The LDC flow involves simulating the motion of an incompressible fluid within a two-dimensional square cavity, where the top lid is assigned a horizontal velocity of u = 1 , while the other walls maintain a no-slip condition. The boundary conditions are mathematically formulated as follows:

The velocity predictions of sPINN and HB-PINN are shown in Fig.42. Due to the simpler geometric configuration of the model compared to the cases discussed in the main text, the improvement in accuracy is less pronounced than that observed in complex-boundary models. However, as evidenced by the residual results in Fig.43, HB-PINN exhibits significantly lower errors near boundaries. This finding aligns with the conclusions drawn in Appendix G, demonstrating that HB-PINN consistently achieves higher precision in boundary regions across diverse models. More detailed error metrics are summarized in the 'LDC' section of Table 6.

t=0.25

t=0.5

t=0.75

t=1

1.0

(a)

8°0

9°0

Temperature

(b)

0.4

0.2

口

(c)

口

口

0'0

t=0.25

t=0.5

t=0.75

t=1

0.14

0.12

(a)

减

0.10

Temperature

0.08

0.06

0.04

(b)

0.02

00°0

SPINN

HB-PINN(ours)

CFD(GT)

1.00

0.75

0.50

0.25

0.00

SPINN

HB-PINN(ours)

0.25

0.20

0.15

0.10

0.05

0.00

K Training and Inference Times

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces Hybrid Boundary Physics-Informed Neural Networks (HB-PINN), a novel approach to solving Navier-Stokes equations (NSEs) with complex boundary conditions. The method addresses the limitations of traditional PINNs, which struggle to balance boundary condition losses and PDE residual losses, especially in geometrically complex domains. HB-PINN achieves lower errors compared to baseline PINN approaches (e.g., sPINN, hPINN, MFN-PINN, XPINN, SA-PINN, PirateNet). Extensive experiments and ablation studies demonstrate its robustness and accuracy in handling complex boundary conditions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n1. Decoupling subnetworks (NP, ND, NH) effectively handles complex boundaries. The distance metric network (ND) is a useful solution for handling complex geometries, ensuring smooth transitions near boundaries.\n2. Error reductions compared to baseline methods. Detailed residual analyses and comparison plots provide clear evidence.\n3. The methodology is extended to transient flows, high Reynolds number scenarios, and other PDEs (e.g., heat conduction). The results indicate potential applicability to a wide range of fluid dynamics problems and other scientific domains.\n\nWeaknesses\n1. Empirical Hyperparameter Selection:\nThe critical parameter α in the distance metric network and the number of pre-training epochs for NP are tuned empirically. This lack of systematic optimization may limit reproducibility and scalability to other problems. The paper does not provide guidance on how to set these parameters for new applications.\n2. Computational Cost:\nWhile the paper demonstrates improved accuracy, it does not discuss the computational overhead of training three separate subnetworks.\n3. The paper primarily focuses on toy problems but does not discuss the practical implications and limitations of HB-PINN for real-world engineering problems (e.g., industrial CFD applications).\n\nQUESTIONS:\n1. How sensitive is the performance of HB-PINN to the choice of α in the distance metric network? \n2. How does the training time and resource consumption of HB-PINN compare to baseline methods?\n3. Can the HB-PINN framework be extended to 3D Navier-Stokes equations?\n4. How does HB-PINN perform for geometries with moving boundaries or time-dependent boundary conditions?\n5. Have you explored the application of HB-PINN to industrial-scale problems (e.g., aerodynamics, weather modeling)?\n6. How does HB-PINN compare to traditional CFD solvers in terms of accuracy and efficiency for such problems?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-28"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nA kind of mixture model is proposed for dealing with complex boundary conditions with PINNs. The idea is to (softly) switch between two networks: one predominantly for achieving boundary conditions and another for the inner domain solution. The method is examined with multiple different flows.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n- The motivation to separately deal with the boundary and inner regions is clearly presented.\n- The proposed method is compared with multiple baseline methods.\n\n### Weaknesses\n\n**(1)**\nDespite the (ostensible) resemblance of the proposed method to hPINN by Lu et al. (2021) [21], I do not think it is discussed well.\nThe design choices different from hPINN should be elaborated on more, with the intentions behind them explained.\nFor example, why $\\mathcal{N}\\_\\mathcal{P}$ is used instead of the original boundary condition function? Why $\\mathcal{N}_\\mathcal{D}$ is necessary? etc.\n\n**(2)**\nThe motivation to use $\\mathcal{N}\\_\\mathcal{D}$ is particularly unclear.\nWhat is the point of learning $\\mathcal{N}_\\mathcal{D}$ when you can compute $\\hat{\\mathcal{D}}$?\nWhy not using $\\hat{\\mathcal{D}}$ in Eq. (3) directly?\n\n**(3)**\nIn the loss function for the particular solution net $\\mathcal{N}\\_\\mathcal{P}$, the motivation to have non-zero $\\lambda_1$ is unclear. Setting much smaller $\\lambda_1$ than $\\lambda_2$ and $\\lambda_3$ is proposed, which itself makes sense, but why $\\lambda_1>0$ is preferable? Is there any ablation study of setting $\\lambda_1=0$?\n\n**(4)**\nLine 172:\n\n> hPINN method in this study selectively relaxes the enforcement of boundary conditions\n\nThe exact setup of the hPINN here is unclear. Please elaborate on the way the BC was relaxed.\n\n**(5)**\nPerhaps related to (4) above, I am not sure if an important baseline (or ablated) method is examined. The proposed method should be compared with a variant of hPINN, where the $\\ell$ function of hPINN is replaced with the proposed $\\hat{\\mathcal{D}}$ function (with the other components being the same with the original hPINN; i.e., no $\\mathcal{N}_\\mathcal{P}$, and no $\\mathcal{N}\\_\\mathcal{D}$). Is there any such baseline or ablated method? \n\n**(6)**\nThe details of the other baseline methods are missing as well. Please clarify the configurations of all the baseline methods.\n\nQUESTIONS:\nPoints **(2)**, **(3)**, and **(5)** in the \"Strengths And Weaknesses\" section  are my questions particularly affecting the evaluation.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a \"Hybrid Boundary Physics-Informed Neural Network\" (HB-PINN) to solve Navier-Stokes equations (NSE) with complex boundaries. The method decomposes the solution into a particular solution that satisfies the boundary conditions and a homogeneous solution that satisfies the PDE. This is achieved using a composite architecture of three subnetworks: a \"Particular Solution Network\" (`N_P`) to approximate the boundary values, a \"Distance Metric Network\" (`N_D`) to create a boundary-aware mask, and a \"Primary Network\" (`N_H`) to solve the PDE in the interior. The `N_P` and `N_D` networks are pre-trained and then frozen, while the larger `N_H` network is trained to minimize the PDE residual. The authors test their method on several 2D fluid flow problems, such as flow around a cylinder and flow in an obstructed cavity, and report state-of-the-art accuracy compared to several existing PINN variants.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths:**\n\n1.  **Clear Motivation:** The paper effectively targets the well-known challenge of balancing PDE and boundary losses in PINNs, providing a strong motivation for the proposed method.\n2.  **Thorough Experiments:** The method is validated on multiple challenging fluid dynamics benchmarks, with comprehensive comparisons against a wide array of existing PINN variants. The appendices provide extensive details, enhancing reproducibility.\n\n**Weaknesses:**\n\n1.  **Limited Originality:** The core method relies on the classical solution decomposition technique (`q = P + D*H`) to enforce hard boundary constraints, which is a well-established paradigm in the PINN literature. Using neural networks to learn the `P` and `D` components is an implementation detail rather than a fundamental conceptual advance, making the contribution feel incremental.\n2.  **Excessive Complexity:** The proposed architecture, involving nine separate networks and a multi-stage, multi-hyperparameter training process, is overly complex and inelegant. This complexity hinders its practicality and scalability compared to simpler hard-constraint methods that achieve similar goals.\n3.  **Redundant Components:** The paper fails to convincingly justify the need for both a \"Particular Solution Network\" (`N_P`) and a \"Distance Metric Network\" (`N_D`). A single, well-formulated distance function is often sufficient to enforce boundary conditions in similar frameworks. The inclusion of the `N_P` network appears redundant and adds unnecessary complexity without a clear, demonstrated advantage over simpler alternatives.\n\nQUESTIONS:\n1.  **On Originality:** The core idea of solution decomposition (`q = P + D*H`) is a well-established concept for enforcing hard constraints in PINNs. Could you please clarify what you view as the primary conceptual novelty of HB-PINN beyond using neural networks to approximate the `P` and `D` functions, which seems like a direct, albeit complex, implementation of this known principle?\n2.  **On Architectural Complexity:** The proposed method uses nine separate DNNs and a multi-stage training process. This is significantly more complex than most standard PINN frameworks. Have you considered a simpler architecture? For example, could a single network be used to output both the solution and an auxiliary distance estimate, or could the `N_P` and `N_D` functions be combined? What is the justification for this high level of architectural complexity?\n3.  **On the Necessity of the `N_P` network:** The `N_D` network, when multiplied by `N_H`, already enforces a zero value at the boundary, which is a common way to satisfy homogeneous boundary conditions. The `N_P` network is introduced to handle non-homogeneous boundary values. However, this can also be accomplished with a single term, e.g., `q(x) = BC(x) + D(x)H(x)`, where `BC(x)` is a simple function (or network) that matches the boundary conditions. Could you explain why the separate, pre-trained `N_P` network, which is weakly trained on the PDE, is superior to simpler, more standard ways of enforcing non-homogeneous boundary conditions? A direct comparison to such a simpler baseline would be very informative.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose Hybrid Boundary PINNs which combines a pretrained boundary condition network along with a primary network which focuses on the inner domain points. A distance metric network is used to enhance the predictions at the boundaries by tuning down the contributions of the pretrained network at the inner points and primary network at the boundary points respectively. This method has been used to solve Navier stokes equations with complex boundaries. Steady state flow past a 2D cylinder, 2D blocked cavity flow with a segmented inlet and a transient flow scenario were considered as examples to demonstrate the efficacy of the proposed framework. The proposed method has been compared against many PINN variants such as soft-constrained PINN (sPINN), hard-constrained PINN (hPINN),  self-adaptive PINN (SA-PINN), modified Fourier network PINN (MFN-PINN), extended-PINN (XPINNs), PirateNets.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n- The authors have provided a very thorough review of related works and have chosen the latest variants of PINNs relevant to the problem being solved.\n- The methodology has been clearly presented without any confusion and nthe details of hyperparameters chosen for each case have also been presented in the appendices without fail. \n- Detailed results and discussion are presented in a clear manner with appendices subsequently elaborating and justifying their findings. \n\n**Weaknesses**\n- The model could have outshined if the case studies considered were transient in nature rather than steady state. Most real life use cases of incompressible flows come from unsteady flows involving different vortex dynamical phenomenon like vortex shedding, vortex merging, etc. \n- Only a single variant of the distance metric network is considered, radial basis functions, gaussian kernels, indicator functions, etc could also be potential options for these distance metric networks which can be explored to study the difficulty of trainability of PINN model. \n- While only contours are presented for the predictions, sectional profiles of velocity and pressure for each case would be more informative. \n- The ansatz based PINN models for hard constraining is not very new. While the authors have cited Sun et al. (2020)'s work [paper no. 11 in the references] on surrogate modelling in the related work section, they have not highlighted how their distance metric network is different from Sun et al. or compared with their work.\n\nQUESTIONS:\n- The model could have outshined if the case studies considered were transient in nature rather than steady state. Most real life use cases of incompressible flows come from unsteady flows involving different vortex dynamical phenomenon like vortex shedding, vortex merging, etc. **Request the authors to consider testing against a transient vortex dominated flow situation.**\n- Only a single variant of the distance metric network is considered, radial basis functions, gaussian kernels, indicator functions, etc could also be potential options for these distance metric networks which can be explored to study the difficulty of trainability of PINN model. **Request the authors to comment on the applicability of the other variants and also the superiority of using a power law.**\n- While only contours are presented for the predictions, have the authors looked at sectional profiles of velocity and pressure for each case which would be more informative? **Especially provide sectional profiles and temporal history (for transient cases) where the flow-field gradients are strong. Request the authors to provide them for various velocity components and pressure.** \n- The ansatz based PINN models for hard constraining is not very new. While the authors have cited Sun et al. (2020)'s work [paper no. 11 in the references] on surrogate modelling in the related work section, they have not highlighted how their distance metric network is different from Sun et al. or compared with their work. But the authors have proposed a distance metric network for hard constraining in the case of complex boundaries which is a relatively novel power law function. \n- But the major weakness is that in the study, time evolving problems / unsteady problems are not considered in detail or given equal/higher importance. Unsteady problems if solved, then steady state problems are easier, but the reverse is not true. **Have the authors considered flow past a cylinder at Re > 100 where vortex shedding happens?**\n- **Computational cost**: Nowhere is the training vs inference costs in comparison to CFD are presented given that the authors are solving a forward problem. **This is crucial performance related information which needs to be provided comparing with all the other PINN variants as well. Please provide a table detailing the computational cost/ efficiency of the proposed models**\n\nLIMITATIONS:\nThe key limitation here is that the distance metric function variants need to be evaluated. Especially since the ansatz form for enforcing hard constraints is not new, comparison with existing literature would be useful to frame a stronger case for the proposed method. If not, atleast the limitations of the network would be more clear when such comparison is carried out.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-11"}