Hybrid Boundary Physics-Informed Neural Networks for Solving Navier-Stokes Equations with Complex Boundary
Authors: Chuyu Zhou
Tianyu Li
Chenxi Lan
Rongyu Du
Guoguo Xin
Wei Li
Guoqing Wang
Xun Liu
Hangzhou Yang
Venue: NeurIPS 2025 poster ()
Keywords: Physics-informed neural networks
Navier-Stokes equations
complex boundary conditions
Primary area: machine_learning_for_sciences

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
Chuyu Zhou"! Tianyu Li Chenxi Lan², Rongyu Du Guoguo Xin Hangzhou Yang', Guoqing
Wang2*+, Xun Liu', Wei Li 1+++
'School of Physics, Northwest University, Xi'an an 710127, China
SSchoo of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 61 1731, China
DDepartment of Imaging Technology, Beijing Institute of Space Mechanics and Electricity, Beijing 100094, China
"School of Information Science & Technology, Northwest University, Xi'an 710127, China Donghai Laboratory, Zhoushan, Zhejiang 316000, China
"xinguo@nwu. edu. qwang04200hotmaall com wei wei.1i.bje13..com com
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 nism. The HB-PINN method features 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 seg- mented inlet. The proposed method achieves state-of-the-art (SOTA) performance on these benchmark scenarios, demonstrating significantly improved accuracy over existing PINN-based approaches.
mecha-
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 al. (I]. 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 33, have emerged notable class of surrogate models for computational fluid dynamics.
PINN represent 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, requirement
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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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 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 61. A notable limitation of this loss formulation lies in its frequent failure to ensure simultancous 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 9].
Here, to address the difficulty in balancing boundary condition losses and equation losses in complex boundary flow field problems, 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-orient 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.
distance function network, this method
The main contributions of this work are as follows:
1. We propose a new PINN approach named HB-PINN that are embedded the boundary conditions solve flow fields in regions with irregular obstructing structures.
2. A power function is proposed to refine the distance metric for handling complex boundary conditions, thereby improving prediction accuracy.
3. Extensive experimental validations on NSE with complex boundary conditions such as the 2D cylinder wake problem and the 2D obstructed cavity flow with segmented demonttemott 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. The methods have been extended to diverse domains, including: fluid mechanics [10, [1]. medical applications [1, heat transfer[[4] 15), and materials science[[6] 17.. Furthermore, multiple PINN variants have been developed based on PINN, such as: Modified Fourier Network PINN (MFN-PINN) hp- VPINN[[9), Conservative PINN (CPINN) 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. pivotal development by Lu al. (2021) 21 proposes a iestance-informet ansatz that provably satisfies boundary conditions through analytically constructed solutions. This architecture enables exclusive optimization of PDE residuals during training. SA-PINN 22 proposes 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-informedd decomposition framework for solving nonlinear PDE on arbitrarily complex geometries. Although these methods show significant improvements 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 (MFN- PINN) to solve flow field problems with complex boundaries. demonstrating its superiority through comparisons with Multi-Layer Perceptron-base PINN (MLP-PINN). Sukumar et al. (25) proposed an R-function-based distance function construction method for enforcing boundary conditions PINN, which significantly improves convenience in first-order derivative-dominates problems. However, the complex boundary analytical distance functions (ADF) constructed by R-functions are not natural functions.

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To address the loss term conflicts induced by complex boundary conditions, this work decouples the boundary constraints into two fublffunciios: particular solution function that strictly enforees 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:
du + (u V)u+
-V u - 0.
V uu=00 O, at
(I)
In the equations, u. and denote the velocity vector, fluid pressure, density and dynamic viscosity coefficients, respectively. For incompressible flows, P and v 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 The framework consists of three subnetworks: Vp.Wp. and NH-- The Wo subnetwork (Particular Solution Network) is trained to satisfy boundary conditions. while the Ni subnetwork (Distance Metric Network) learns boundary- condition-awer weights by encoding the distance from interior points to domain boundaries. The NN subnetwork (Primary Network) is dedicated resolving the governing PDE. During the final training phase, only the parameters of Nu are optimized, whereas Np and No remain fixed as pre-trained components. This design effectively addresses challenges involving hybrid interior-exterror boundary configurations. Therefore, we term this methodology Hybrid Boundary Networks.
Informed Neural
In our model, the constraints on the variables are formulated as follows:
(xx,)) t) = P,(xx (x,,t) Hx, t),
here, q(-) denote the physical quantities of interest (e.g., and in 2D flow model); Pa represents an additional solution function that satisfies the boundary conditions; D, is the distance function; H.g is the output of the primary network. The distance function D, takes value of 0 at the domain boundaries and rapidly increases to as moves away from the boundaries. Functionally, Dg acts as weight for Hg. modulating its influence across the computational domain.
Under this formulation, the distance function Dy enforces strict boundary condition compliance by evaluating to zero at domain boundaries while smoothly transitioning to Hyg which satisfies the governing equations in interior regions. When boundary conditions and geometric configurations are simple, both P, and DD, can be analytically expressed. However, analytical expressions for P, and D2 are generally infeasible for complex geometries. Consequently, we leverage three dedicated deep neural networks (DNN). the previously defined Np, Ny. and Nx to parameterize these components. The composite solution for 2D incompressible flows is thus constructed as:
Ng(x,t) No, (x, NN. (2,t).
Sub-network N
The subnetwork Nyp adopts the same architecture conventional PINN, with outputs comprising the velocity components u, v. and pressure p. The loss function for training Np is formulated by integrating residuals of the NSE, boundary conditions (BC), and initial conditions (IC), expressed as:
L =>D AeLic + XaCsc.
(4)
3

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a.
P D211
f--
(000)) (n Viw (Uvyy
Pretabeed
Parionas mostonatooo Av
Duraace mesic sahwak No
Co
Figure 1: The architecture the proposed method.
where
Nac
Lgc
|u(x,,t) usc(2,, t,)
(5)
NBC
1 Nic Lic NIc
lu(x,,t)) (oc(2,,t)))
(6)
and
Nyue
Ou Ot
LADE
Σ aml
ull2+
+ (u V)u+ Vp vvau
NPDE
Here, A, = 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 Np subnetwork prioritizes boundary and initial condition enforcement by assigning X2, A3 > λ (e.g., A = d2. A 1000), enabling preliminary PDE training while strictly satisfying boundary constraints.
By decoupling the training objectives, the subnetwork Np focuses exclusively 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 conventional PINN, while delegating full PDE resolution to the primary network NH-
Sub-network
The ND subnetwork is shallow DNN designed construct distance functions. For each quantity of interest, we introduce distance metric network Nv,, To train this network, sampling points (x, !) are firstly collected from the space-time domain Ω x [o, 7]. The network is then supervised to approximate the signed distance function D. t) between these points and the boundary associated with g, formulated as:
D. min (distance to the spatiotemporal boundary of q).
(8)

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In order to ensure the network outputs approach 0 near domain boundaries and rapidly increase to in regions far from boundaries, while maintaining dominance of the primary network NH, predictions in interior regions, we employ a power-law function f f(2g) derived from the distance metric Dj as the training labels for the subnetwork No., The power-law function f(2)) is defined as follows:
f(D,) 1- (1-11-0))) D./ max(
The parameter a is a positive value controlling the growth rate the function. The curves exhibit steeper gradients near boundaries and smoother transitions in regions distant from boundaries. Larger a 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 a values may lead to erroneous predictions of near- boundayy 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, a can tuned to maintain the steepness of the power-law function within reasonable range. The loss function for the distance metric network is defined as:
ND Σ 11 - (f 1?2 LD, ND4 1=1
(10)
Here, Dy represents the network's output, supervised by the training labels f(D,). This approach ensures robust training of ND, even under highly complex boundary conditions, while simultaneously enhancing the accuracy of the primary network's training outcomes.
Sub-network N
The Nu subnetwork also employs DNN architecture but with larger scale compared to the Ni and Np subnetworks. Through pre-training of the Np and N₂ subnetworks, boundary conditions strictly enforced, allowing Nh to focus solely minimizing the governing equation residuals. For the 2D incompressible NSE, the loss function the Nu subnetwork is defined as:
NIDE
1 -V vV%al Ōú + ú Vû+ a
CH
I 2
+
(1I)
NPDR
0
While this formulation shares the same structure as Eq 7 the variables ú and و are derived from the modified outputs of Eq This training strategy enables the N/ subnetwork 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 wwo-dimensional incompressible flow cases are investigated, including two steady state cases and one transient case, 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 At 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- PINNN 18 24, extended PINN (XPINN) 23 self-adaptive
PINN (SA-PINN) 22 and PirateNets 26 Notably, the hPINN, proposed by Lu al. (1]] exhibits erratic behavior in scenarios with complex boundary conditions, often failing to produce reliable results for comparative analysis. To ensure comparability results, 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 (0 < x < 0.4, y 0& = 1). The detailed rationale for relaxing boundary conditions is provided in Appendix D. Additionally, to
5

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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 cylinder, steady-state flow in segmented inlet with obstructed square cavity, and transient flow in segmented inlet with an obstructed square cavity.
4.1 Experimental Settings
As illustrated in Fig. I the proposed HB-PINN takes space-time coordinates as inputs and outputs velocity components (u, ?) and pressure p. In the implementation phase, the training process employs a distributed architecture: each subnetwork (Np, Ny. and NH) utilizes three independent DNN dedicated to variables v, and resulting in total of nine distinct DNN. After obtaining separate predictions for u, v, and p, these outputs are integrated through Eq. 3 reconstruct the composite solution. For detailed training protocols refer Appendix B.
In the models discussed this study, the NSE are non-dimmensionalized 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.
X
No
5.00
025
Figure 2: The trained boundary prediction for the Case 1. Np, represents the result of the particular solution network for u, while NNp., ND., and No, respectively represent the results of the distance metric network for u, v, and p.
SPNNN
hPINN
MFN-7NN
X-PINN
15
15
HB.R PUNss
CFDIGG
05
Figure Velocity distributions for the flow around cylinder from different methods.
4.2 Case Two-dimensional flow around a cylinder
The flow past circular cylinder represents a canonical benchmark in fluid mechanics(see Appendix A.1 for detailed geometric specifications). The boundary conditions are specified as follows:
1, I = 0, 0 S ny Following Eq the loss function for the Nop subnetwork in the proposed model is formulated as:
= 0, others; p= 5, 0 5
Lp =>= | a|| + |laaatt + (ū V a 1/DV8-
~V20ll2 + Xa -14
+ 1lpp pa p
Here, Ω denotes the interior of the geometric region, while a denotes the boundary. The loss weight A for the equation part is set 1, and the loss weight A for the boundary part is set 1000, with the aim of making the network focus more the training of boundary conditions.

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The outputs of the Mp and No networks are shown in Fig 2 The final modification to the variable result as follows:
à = -P. (z, t) +ND, (z, NN. (z, t), = ND. (). (a, and .ND, (z, t) NNH (z, t).
Since both v and are set to O at the boundary. there is no need to configure the boundary results separately; they simply constrained by the distance function.
In the case of 2D cylindrical pipe flow model, the qualitative results from sPINN, hPINN, MFN- PINN, XPINN, SA-PINN, PirateNet. HB-PINN are shown in Fig The residuals of these methods compared with CFD results are shown in Fig More detailed error metrics are summarized in the "Casel" section of Table
SPINN
NPINN
MEN-PINN
1.00
X-PINN
SA-PIN
ProteNet
0.5
HI-PINNlours)
Figure 4: The residuals of sPINN, hPINN, MFN-PINN, XPINN, SA-PINN, PirateNet, and our HB-PINN compared to the GT in Case
4.3 Case 2: steady-state flow in segmented inlet with an obstructed square cavity
The model incorporates two rectangular obstructions positioned along the top and bottom walls of square cavity, with segmented inlet configuration (see Appendix A.2 for detailed geometric specifications). The boundary conditions are configured as follows:
u 0.5, x y E [0, 0.2] (0.4, [0.8, 1]; == 0, others;
V = O, on M.;
0, x = 1, E [0.8,
The output results of the boundary condition solution function network related to u and the distance function network for u, and p shown in Fig
No.
N
No.
HHH
6550
625
Figure 5:
The trained boundary prediction for Case NN, represents the result of the particular solution network for while N,,,, and ND, respectively represent the results of the distance metric network for u, v and p.
The velocity distributions for case 2 from sPINN, hPINN, MEN-PINN, SA-PINN, XPINN, PirateNet and our HB-PINN are shown in Fig 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 metries are
summarized in the "Case2" section of [
Based 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 lower accuracy within the domain. Under the mD approach, while internal errors are reduced during training, the enforcement of
7

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SPINN
hPINN
MFN FINN
XPINN
2.00
54. PINN
HrateWet
HB PINNicursi
(FD(GT)
1.00
675
Figure 6: Comparison velocity distributions for Case 2 across different methods.
hPINN
HB-PINN(ours)
XPINN
SA.PINN
HrateNet
H
0.25
000
Figure 7: The residuals of sPINN, hPINN, MFN-PINN, XPINN, SA-PINN, PirateNet, and our HB-PINN compared to the GT in Case
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
For a more detailed error analysis, please refer to Table Appendix
II-PPINNious!
20
0.8
66
8.6
0.4
B4
64
84
CFDIGT]
0.
02
42
0.2
as
25
0335 250 0.75
0.50 675
6.5 035
635 030 615
mP Error
mD Error
hPINN Error
HB PIN Emror
0.75
0.8
0.
ca-
425
Figure 8: Comparison of velocity results and residuals between the mP and mD methods in Case 2.
4.4 Case 3: transient flow in segmented inlet with an obstructed square cavity
The difference between this model and case 2 lies the consideration of the transient situation, which further increases the complexity of the boundary conditions. The boundary conditions are set

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as follows: the inlet normal velocity 0.5, the outlet static pressure is 0, and the velocity at other boundaries is set to satisfying the no-slip condition. The initial conditions are set with the veloc- ity components u, o and the pressure p all equal to O. Relative to Case 2, the addition of initial conditions:
Table 1: Comparative analysis error metrics across three Amnns.Among the baseline methods, PirateNet achieves the highest accuracy, while HB-PINN represents our proposed methodology.
Error Metrics
Case
Method
MSF(C)
MAE())
Relative I.2(L)
SPINN
0.520 78
0.642 16
0.668 05
0, v 0.p 0,
hPINN
0.583 17
0.527 29
0.707 12
MEN-PINN
0.423 73
0.574
0.213 63
The network structure parameters cho- Case sen for the transient case are the same as those for case 2. Fig. shows the veloc- ity results obtained using different meth- ods / 1. In the transient case, due the increased complexity of the bound- ary conditions, the accuracy of the other methods significantly decreases, while HB-PINN maintains a high level of ac- curacy. The velocity results and the error compared to the CFD method are shown in Fig. I0 These results demonstrate the versatility of HB-PINN both steady- Case 3 state and transient situations.
XPINN
1.119
0.909 27
0.979 36
SA-PINN
0.053 25
0.189
0.21366
PirateNet
0.00763
0.06689
0,08089
HIB-PNN
0.00433
0.02012
0.06093
PINN
0.008
0.051 73
0.200 36
hPINN
0.040 60
0.112 42
0.451 10
MFN-PINN
0.116 26
0.211 86
0.763 37
Case 2
XPINN
0.021 49
0.083
0.328 24
SA-PINN
0.005
0.045
0.16495
PirateNet
0.00186
0.02357
0.09679
HB-PINN
0.00088
0.01888
0.06655
SPINN
0.12400
0.218
0.788 19
hPINN
0.155
0.239 51
0.882 88
MFN-PINN
0.105 43
0.201 50
0.726 78
XPINN
0.041 49
0.120
0.455 92
SA-PINN
0.106
0.196 42
0.729 15
PirateNet
0.03202
0.10682
0.40054
HB- BINN
0.00825
0.04870
0.20332
XPINN
125
SA-PENN
MrateNet
H6-PINNicurs)
1.00
H H
0.00
Figure 9: Comparison of velocity distributions at t for different methods Case 3.
4.5 Result Discussion
The quantitative results of the error metries between CFD calculation and the predictions from sPINN, hPINN, MFN-PINN, SA-PINN, XPINN, PirateNet and HB HB-PINN for the three cases are presented 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 emoor in solving such problems. The training results of SA-PINN and XPINN show improvements accuracy; however, significant errors persist in the cases with more complex boundary conditions.
Table I Combined with the qualitative results shown in Fig Fig 4667

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SPINN
HPINN
MEN-PINN
PER
125
HB-PINN(ours)
100
XPINN
SA-PINN
AratoNet
PPP
0.50
0.25
Figure 10: The residuals of sPINN, hPINN, MEN-PINN, XPINN, SA-PINN, PirateNet, and HB-PINN compared to the GT in Case
5
Limitations
The optimal values of critical parameters in the methodology. such as the a 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.
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 primary network focused on inner domain points and distance metric network that enhances predictions at the boundaries, ensuring accurate solutions for both boundary and interior regions. This makes the approach applicable to adaptable to more complex fluid dynamics models. Through tests steady-state two-dimensional flow around cylinder, steady-state segmented inlet with obstructed square cavity flow, and transient segmented inlet with obstructed square cavity flow, method outperforms typical PINN approaches large margin, establishing itself as a benchmark for solving NSEs with complex boundaries. Additional analysis provided in the appendices to further illustrate the effectiveness of method.
Acknowledgements
The authors would like to acknowledge the financial support received from Donghai Laboratory (2024SSYS0091) Open Fund Beijing Key Laboratory of Advanced Optical Remote Sensing Tech- nology (AORS202408), National Natural Science Foundation of China (12304346). Development Program of the Department of Science and Technology Shaanxi Province and Haina Plan of Zhouchuang Future from Mei Lin.
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References
[1] B. R. Munson, D. Young, T. Okiishi, Sharon Hollars, and Katie Johnson. Fundamentals of Fluid Mechanics, page 53-84. WORLD SCIENTIFIC, Apr 2006.
[2] A. McLay. Computational fluid dynamics: the basics with applications. The Aeronautical Journal, 100(998):365-355, 1996.
[3] Yann LeCun, Yoshua Bengio, and Geoffrey Hinton. Deep learning. Nature, 521((553):443 444, May 2015.
[4] M. Raissi, P. Perdikaris, and G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, Feb 2019.
[5] Tsung- Ych Hsieh and Tsung-Hui Huang. A multiscale stabilized physics informed neural networks with weakly imposed boundary conditions transfer learning method for modeling advection dominated flow. Engineering with Computers, May 2024.
[6] M. Lai, Yongcun Song, Xiaoming Yuan, Hangrui Yue, and Tianyou Zeng. The hard-constraint pinns for interface optimal control problems. arXiv.org, 2023.
[7] Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific machine learning through physics-informed neural networks: Where are and what's next. Journal of Scientific Computing, 92(3), Jul 2022.
[8] Jinshuai Bai, Timon Rabezuk, Ashish Gupta, Laith Alzubaidi, and Yuantong Gu. A physics- informed neural network technique based on a modified loss function for computational 2d and 3d solid mechanics. Computational Mechanics, Nov 2022.
[9] Rafael Bischof and M. Kraus. Multi-objective loss balancing for physics-informed deep learning. arXiv.org, 2021.
[10] Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Karniadakis. Physics-informed neural networks (pinns) for fluid mechanics: review. Acta Mechanica Sinica, 37(12):1727-177 1738, Dec 2021.
[11] Luning Sun, Han Gao, Shaowu Pan, and Xia Wang. Surrogate modeling for fluid flows based chostan-cos deep learning without simulation data. Computer Methods in Applied Mechanics and Engineering, 361:1 12732,, 2020.
[12] Francisco Sahli Costabal, Yibo Yang. Paris Perdikaris, Daniel E Hurtado, and Ellen Kuhl. Physics-informed neural networks for cardiac activation mapping. Frontiers in Physics, 8:42, 2020.
[13] Mohammad Sarabian, Hessam Babaee, and Kaveh Laksari. Physics-informed neural networks for brain hemodynamic predictions using medical imaging. IEEE transactions medical imaging, 41(9):2285-255 2303, 2022.
[14] Shengze Cai, Zhicheng Wang, Sifan Wang, Paris Perdikaris, and George Em Karniadakis. Physics-informed neural networks for heat transfer problems. Journal of Heat Transfer, 143(6): 060801, 2021.
[15] Navid Zobeiry and Keith D Humfeld. A physics-informed machine learning approach for solving heat transfer equation in advanced manufacturing and engineering applications. Engineering Applications of Artificial Intelligence, 101:104232, 2021.
[16] Enrui Zhang, Ming Dao, George Em Karniadakis, and Subra Suresh. Analyses of internal structures and defects in materials using physics-informed neural networks. Science advances, 8(7)xeabk0644, 2022.
[17] Khemraj Shukla, Ameya Jagtap, James L Blackshire, Daniel Sparkman, and George Em Karniadakis. A physics-ifformed neural network for quantifying the microstructural properties of polycrystalline nickel using ultrasound data: promising approach for solving inverse problems. IEEE Signal Processing Magazine, 39(1):68-77, 2021.

---

[18] Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Keil., Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng. Fourier features let networks learn high frequency functions in low dimensional domains. Advances in neural information processing systems, 33:7537-7547, 2020.
[19] Ehsan Kharazmi, Zhongqiang Zhang, and George Em Kamiaadaki.. hp-vpinns: Variational physics-informed neural networks with domain decomposition. Computer Methods in Applied Mechanics and Engineering. 374: 13547, 2021.
[20] Ameya D Jagtap, Ehsan Kharazmi, and George Em Karniadakis. Conservative physics-informed neural networks on discrete domains for conservation laws: Applications forward and inverse problems.
Computer Methods in Applied Mechanics and Engineering, 365: 13028, 2020.
[21] Lu Lu, Raphaël Pestourie, Wenjie Yao, Zhicheng Wang, Francese Verdugo, and Steven G. Johnson. Physics-informed neural networks with hard constraints for inverse design. SIAM Journal on Scientific Computing. 43(6):B1 (00--BB110-BBBB 111-111511111 132,, Jan 2021.
[22] Levi D. McClenny and Ulisses M. Braga-Neto. Self-adaptive physics-informed neural networks. Journal of Computational Physics, 474: 2023. ISSN 0021-9991.
[23] Ameya D. Jagtap and George Em Karniadakis. Extended physics-informed neural networks (xpinns): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations. Communications in Computational Physics, 28(5),
2020.
[24] Tommaso Botarelli, Marco Fanfani, Paolo Nesi, and Lorenzo Pinelli. Using physics-informed neural networks for solving navier-stokes equations in fluid dynamic complex scenarios. Engi- neering Applications of Artificial Intelligence, 148:1 10347, 2025. ISSN 0952-1976.
[25] N. Sukumar and Ankit Srivastava. Exact imposition of boundary conditions with distance functions in physics-infomed deep neural networks. Computer Methods in Applied Mechanics and Engineering, 389:1 14333, 2022. ISSN 0045-7825.
[26] Sifan Wang. Bowen Li, Yuhan Chen, and Paris Perdikaris. Piratenets: Physics-infrrmed informed deep learning with residual adaptive networks. Journal of Machine Learning Research, 25(402): 1-51, 2024.
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A
Supplementary Geometric Details
A.1 Case1:Flow around a Cylinder
Figure 11: Scheme of the two-dimensional flow around cylinder.
Case corresponds to the canonical flow past a circular cylinder. As schematically depicted in Fig Following non-dimensioonalizationn the computational domain is configured with length of 5 and a width 1. A circular obstruction with diameter = 0.2 is centered at (æ, y) = (0.5, 0.5). The boundary conditions defined as follows:
Inlet (Left Boundary): Horizontal inflow velocity 1, 0.
Outlet (Right Boundary): Static pressure fixed at = 0.
Top/Bottom Boundaries: No-slip walls (u = 0).
This configuration corresponds to Reynolds number Re 100, with the dimensionless density The dynamic viscosity is determined by the Reynolds number formula Re w. where denotes the characteristic velocity and D the characteristic length.
P =
v
A.2 Case2:Steady-Stue Flow in Segmented Inlet with an Obstructed Square Cavity
H
Figure 12: Scheme of the Segmented inlet with obstructed square cavity flow.
The model features two rectangular obstructions positioned the top and bottom walls of square cavity with an edge length of unit. As illustrated in [2g the inlet is divided into three segments, and the obstructions have width of 0.2 units and height of 0.3 units. The boundary conditions defined as follows:
Inlet: Normal inflow velocity u 0.5, v 0.
Outlet: Statie pressure set to p 0.
Other Boundaries: No-slip conditions (u 0) all walls and obstruction surfaces.
In Case 3 of the main text, the geometric configuration remains identical to that Case 2, with the sole distinction lying in the incorporation of transient flow dynamics.
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B
Training Details
The framework is developed on the PyTorch machine learning platform, and training iterations are accelerated using nigh-performance GPU.
B.1
Casel
Subnetwork Np:
Network architecture: Three separate deep neural networks (DNNs) for variables u, v, and each structured as [2] + [64] [1]
Training epochs: 10,000
Activation function: tanh
Optimizer: Adam
Learning rate: 1 x 10-3
Subnetwork Np:
Network architecture: Three separate DNNs for u, v, and each structured as [2] +4 [20] [[]]
Training epochs: 300,000
Activation function: tanh
Optimizer: Adam
Learning rate: Initial learning rate of 1 10-3 with learning rate annealing strategy
10
Subnetwork Nx::
Network architecture: Three separate DNNs for u, v, and p. each structured as [2]+6x [128] +[1]
Training epochs: 300,000
Activation function: tanh
Optimizer: Adam
Learning rate: Initial learning rate of 1 x 10-3 with a learning rate annealing strategy
B.2
Case2
Subnetwork.No Np:
Network architecture: Three separate deep neural networks (DNNs) for variables u, v, and each structured [2] + 6 [64] [1]
Training epochs: 10,000
Activation function: tanh
Optimizer: Adam
Learning rate: x 10-3
Subnetwork Np:
Network architecture: Three separate DNNs for u, v, and 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 learning annealing strategy
5
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Subnetwork Ny::
Network architecture: Three separate DNNs for u, v, and p. each structured as [x [128] +[1]
Training epochs: 500,000
Activation function: tanh
Optimizer: Adam
Learning rate: Initial learning rate of 1 10 with a learning rate annealing strategy
B.3
Case3
Subnetwork Np:
Network architecture: Three separate deep neural networks (DNNs) for variables u, v, and p.
each structured as [3] + 6 x [64] [1]
Training epochs: 10,000
Activation function: tanh
Optimizer: Adam
Learning rate: 1 10-3
Subnetwork Np:
Network architecture: Three separate DNNs for u, v, and p. each structured (3) +4x (64) [[]]
Training epochs: 300,000
Activation function: tanh
Optimizer: Adam
Learning rate: Initial learning rate of 10-3 with learning rate annealing strategy
5
Subnetwork Nz::
Network architecture: Three separate DNNs for v, and p, each structured as (3] +6x [128] [[1]
Training epochs: 500,000
Activation function: tanh
Optimizer: Adam
Learning rate: Initial learning rate of 10-3 with learning rate annealing strategy
B.4 Other PINN methods
Network architecture: Three separate DNNs for u, v, and p. each structured as [2]+6x [188+[[]]
Training epochs: 500,000
Activation function: tanh
Optimizer: Adam
Learning rate: Initial learning rate of x 10-3 with a learning rate annealing strategy
Points per epoch: 20,000 (random sampling from domain)
C
Ablation studies
The three ablation studies follows:
1. Comparative analysis of mP (particular solution network only) and mD (distance metric network only) against hPINN and HB-PIN (IB-INN,, where mP and represent experimental configurations using isolated components of HB-PINN.
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2. Impact assessment of varying pre-training epochs for the particular solution network (Np) on the final accuracy of the primary network.
3. Sensitivity analysis the power-law exponent the distance metric network (Np), which governs the transition steepness of the distance function, to evaluate its influence on solution accuracy.
C.1 mP and mD
In this section, 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 ml) refers using only the distance metric network. The experimental results compared with hPINN and HB-PINN, and velocity predictions are shown in Fig 13
mp
mD
2.00
04
04
1.50
125
hPINN
12
HB-PINN(ours)
14
CFDIGT)
1.00
0.75
0.50
0.5
02
06
06
Figure 13: Comparison of velocity results of the mP and mD methods in Case 2.
mP
mD
00
06
04
02
0.50
64
62
hPINN
HB-PINN(ours)
0.5
04
04
03
0.00
Figure 14: The residuals of mP, mD, hhPhh,NNN... and HB-PINN compared to the GT in Case 2.
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.
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Under the mD approach, while internal errors are reduced during training, the enforcement of boundary conditions is imprecise, which affects the final results. These 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 shown in Fig 144More 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 (Np) trained with shallow deep neural network (DNN) using 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 s output 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 the network's '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 NN on the final training results. The velocity results and their residuals relative to the CFD method are shown in Fig 16
epach
epoch
1.75
3.50
3.25
0.50
0.25
Figure 15: Final velocity results for Nyp trained with 10,000, 30,000, and 50,000 iterations.
10000. 10o00_epach
30000, epoch
$0000_eeood
030
0.05
52
13
Figure 16: Residuals of the final veloeity results compared the CFD method for Npp trained with 10,000, 30,000, and 50,000 iterations.
As evidenced by the error metrics in Figure 15 and the "Epoch for Np" section of Table 2, limiting the training iterations of the particular solution subnetwork (Wp) leads to lower global error within computational domain and more rapid convergence of the primary network (N+). However, this configuration results in reduced precision in boundary regions compared to cases where Wp undergoes extended training. Curigenturrrn.ty... the number of training iterations for Np determined empirically, and further research needed to balance accuracy and convergence speed optimally.

---

C.3 Distance metric network under different
This section evaluates the impact of four critical parameter values 5, 10, 15) in the distance metric network on the predictive performance of the HB-PINN method. Notably, 5 corresponds 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
2.00
CFD(GT)
1.75
1.50
1.25
10
a=
1.00
0.75
0,50
0.25
0.00
Figure 17: Comparison of velocity results between HB-PINN and CFD for a = 3, 5, 10, and 15.
4-3
8 5-5
10
15
K
Figure 18: Residuals between HB-PINN and CFD velocity results for a = 3, 5, 10, and 15.
The parameter governs the steepness of the transition in the distance function near boundaries.
Excessively small a values (e.g., a = 3) result in insufficient sensitivity to boundary constraints, while overly large values (e.g.,
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 metries of the mean squared error (MSE) shown in Fig 18 and the a for ND section of Table These results indicate that both undersized and oversized o values degrade prediction accuracy. However, the selection of currently remains empirically determined, and identifying an optimal 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 Wp is effective when the boundary conditions are simple, However, for complex boundaries, this approach leads to distorted discontinuous outputs within the solution domain, especially near junctions between different boundary types.
Fig 19 presents the results of the boundary particular solution network Npp from the original hPINN, hPINN with partially relaxed boundary constraints, and HB-PINN. can be clearly observed that under the original hPINN framework, the output of the Mp network exhibits irregular solutions within the domain. Since it is trained solely on boundary conditions, internal results are uncon- trollable. This significantly increases the training difficulty of the main network, even hindering
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Table Comparative analysis of error metrics across ablation studies
Error Metrics(L)
Category
Configuration
MSE(L)
MAE(L)
Relative L2(()
hPINN
0.040 60
0.112 42
0.451 10
mp
0.004 84
0.045 89
0.155 75
mP & mD
mD
0.004 09
0.022 69
0.143 29
HB-PINN
0.000 88
0.018 88
0.066 55
10000
0.000 88
0.018 88
0.066 55
Epoch for Np
30000
0.002 34
0.019 07
0.108 33
50000
0.002 48
0.016 44
0.111 69
3
0.004 71
0.025 77
0.153 74
0.000 88
0.018 88
0.066 55
a for N
10
0.001 77
0.023 12
0.094 29
15
0.001 40
0.021 99
0.084 03
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 Table fully demonstrate this point. In Relaxed hPINN, we relaxed the boundary constraints in the regions O < 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 Np with weak equation constraints an alternative solution aimed at more robustly handling complex boundary constraints.
Original hPINN
Relaxed hPINN
HB PINN
25
0.8
0.8
94
03
02
21
Figure 19: Comparative Analysis of Particular Solution Network Results across Original hPINN, Relaxed hPINN, and HB-PINN
Table 3: Loss comparison across different methods at various epochs
Epoch
Original hPINN
Relaxed hPINN
HB-PINN
335 810
47.44
7.129
1000
15 292
29.82
2.517
2000
855 618
30.75
1.797
3000
720 168
28.45
1.588
E
Supplementary details of the transient case (Case 3).
In the main text of Case 3, we present the velocity results at t = for the transient segmented inlet with obstructed square eavity flow model. Att =tt= = the flow can considered have developed to state approaching steady-state results. Fig 20
22 23 display the velocity results at several time
26

---

steps during the development of the transient model, specifically a t= t.3, and t= 0.7, 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 different time steps and their corresponding square relative the CFD method are quantitatively summarized in Table4
SPINN
hPINN
MEN-FINN
XPINN
175
150
SA-FNN
BrateNet
HB-PINNNfuus!
OFDrGT)
425
Figure 20: Comparison of velocity distributions at t 0.1 for different methods in Case 3.
SPINN
APINN
MEN-PINN
XPINN
175
150
SA-PINN
KrateNet
HBBPIINMouus!
OFDrGT)
6.5
425
Figure 21: Comparison of velocity distributions at t 0.3 for different methods in Case 3.
SPINN
hPINN
MEN-PINN
XPINN
175
150
SA-PINN
StateNet
II8-PINMeurs!
oFDrGT)
150
425
Figure 22: Comparison of velocity distributions at t= 0.5 for different methods in Case 3.
27

---

SPINN
hPINN
MFN FINN
XPINN
2.00
S4. PINN
HrateWet
HB PINNicurs]
(FD(GT)
675
Figure 23: Comparison of velocity distributions 0.7 for different methods in Case 3.
SPINN
hPINN
HB-PINN(ours)
100
0.75
XPINN
SA,PINN
PirateNet
025
0.00
Figure 24: Comparison of the residuals relative to the ground truth (GT) for sPINN, hPINN, SA- PINN, XPINN, and our HB-PINE t 0.1 in Case 3.
SPINN
hPINN
MEN-PINN
PER
HB- (I-PINN(ours)
100
XPINN
SA-PINN
PirateNet
0.25
0.00
Figure 25: Comparison of the residuals relative the ground truth (GT) for sPINN, hPINN, SA- PINN, XPINN, and our HB-PINN at t 0.3 in Case 3.
28

---

SPINN
HPINN
MEN-PINN
PEP
125
HB-PINN(ours)
100
XPINN
SA-PINN
ArrtoNett
0.500
0.25
Figure 26: Comparison of the residuals relative the ground truth (GT) for sPINN, hPINN, SA- PINN, XPINN, and our HB-PINN at t = 0.5 in Case 3.
SPINN
hPINN
MEN-PINN
PEP
HB-PIN (O--IINN(ours
100
0.75
PAP
XPINN
PINN
PrateNett
0.00
Figure 27: Comparison of the residuals relative to the ground truth (GT) for sPINN, hPINN, SA- PINN, XPINN, and HB-PINN 0.7 in Case 3.
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. 28 displays the predictive results along the cross-section y 0.5 in Case compared with CFD results, while Fig 29 presents the temporal distribution of predictions the center point (0.5, 0.5) in Case 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.
Figure 28: Comparison of velocity and pressure profiles along the 0.5 cross-section in Case 2.
29

---

Table 4: Comparative analysis of error metries across different methods at multiple time steps
Error Metrics
Time
Method
MSE())
MAE(L)
Relative L2())
SPINN
0.111 93
0.211 71
0.782 71
hPINN
0.16037
0.258 55
0.936 88
MFN-PINN
0.105
0.206 14
0.759 67
t = 0.1
XPINN
0.054 66
0,151 06
0.546 99
SA-PINN
0.07638
0.172 18
0.646 59
PirateNet
0.05228
0.14450
0,53493
HB-PINN
0.00873
0.05070
0.21864
sPINN
0.119
0.216
0.788 15
hPINN
0.15441
0.241 52
0.897 54
t 0.3
MFN-PINN
0.109 33
0.208 13
0.755 23
XPINN
0.041 74
0.122 88
0.466 66
SA-PINN
0.067 36
0,152 83
0.592 80
PirateNet
0.01989
0.09057
0.32215
HB-PINN
0.00580
0.04194
0.17405
sPINN
0.122 18
0.218 31
0.789 21
hPINN
0.155
0.240
0.890
MFN-PINN
0.111 64
0.209 94
0.754 41
f 0.5
XPINN
0.040
0.119
0.454
SA-PINN
0.077 00
0.164 65
0.626 53
PirateNet
0.02027
0.08035
0.32152
HB-PINN
0.00700
0.04895
0.18901
sPINN
0.123 68
0.219 27
0.789 48
hPINN
0.156 07
0.240 33
0.886 85
MEN-PINN
0.11283
0.21061
0.754 05
t t= 0.7
XPINN
0.040 87
0.11990 9D
0.453 83
SA-PINN
0.089 56
0.17947
0.671 81
PirateNet
0,02522
0,09204
0.35652
HB-PINN
0.00805
0.05228
0.20079
SPINN
0.124
0.218
0.788
hPINN
0.155 59
0.239 51
0.882 88
MEN-PINN
0.105
0.201
0.726 78
1 t 1.0
XPINN
0.041 49
0.120 28
0.455 92
SA-PINN
0.106
0.196
0.729
PirateNet
0,03202
0. 00682
0.40054
HB-PINN
0.00825
0.04870
0.20332
Figure 29: Comparison of Temporal Distributions of Velocity and Pressure at (0.5, 0.5) in Case 3.
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
30

---

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
MI PINN
XPINN
2.00
1.75
S4-PIIN
ArateWet
HB PI--IiN
(FDIGT)
675
Figure 30: Comparison of velocity results under different methods for steady-state flow in square cavity with obstructed segmented inlet at Reynolds number He = 500.
sPINN
hPINN
MEN PINN
0.75
HB- PiNu(s
u.00
XPINN
SA-PINN
PirateNet
0.55
Figure 31: Comparison of velocity residuals relative to ground truth (GT) for steady-state flow in a square cavity with obstructed segmented inlet at Re 500, evaluated under different numerical methods.
SPIN
NPINN
MEN-PINN
XPINN
SA-PINN
BrateNet
IIB-PINNcurs!
CTDIGT)
1.00
Figure 32: Comparison of velocity results under different methods for steady-state flow square cavity with obstructed segmented inlet at Reynolds number Re 1000.
31

---

SPINN
HPINN
MEN-PINN
0.75
HB-PINN(ours)
H
0.50
XPINN
SA-PINN
ArrtoNet
Figure 33: Comparison of velocity residuals relative ground truth (GT) for steady flow in square cavity with obstructed segmented inlet at Re = 1000, evaluated under different numerical methods.
SPINN
hPINN
MEN-FINN
XPINN
2.00
SA-PINN
PrateNet
HB- HB-PINNiours!
(FDIGT)
1.00
6.75
Figure 34: Comparison of velocity results under different methods for steady-state flow square cavity with obstructed segmented inlet at Reynolds number Re 2000.
SPINN
hPINN
MEN PINN
0.75
HB- (I--PiN((ours)
0.50
XPINN
SA-PINN
PirateNet
0.25
Figure 35: Comparison of velocity residuals relative to ground truth (GT) for steady-state flow in square cavity with obstructed segmented inlet at Re 2000, evaluated under different numerical methods.
Fig 30 32 34 present the velocity predictions of different methods at Re 500, 1000, and 2000,
respectively, while their residuals relative high-fidelity CFD results are visualized in Fig 30 33 35 As shown in the MSE variation trends across Reynolds numbers (Fig 36), all methods exhibit rising
32

---

prediction errors with increasing Re. However, the proposed HB- HB-PINN consistently achieves the lowest MSE values at every Reynolds number.
Quantitative error metrics under multiple evaluation criteria are tabulated in Table Notably, HB-PINN achieves
wyder-of-magnituee reduction in MSE compared to conventional methods at Re 500 and 1000. Even under the most challenging Re 2000 condition, HB-PINN demonstrates notable accuracy improvement.
Table 5: Comparative analysis of error metrics across different methods under varying Reynolds numbers (Re).
Error Metrics
Re
Method
MSE())
MAE())
Relative I.((L)
SPINN
0.11463 63
0.205 69
0.740 24
hPINN
0.056 82
0.139 10
0.521 16
MFN-PINN
0.126 93
0.221 63
0.778 92
500
XPINN
0.084 25
0.179 04
0.634 62
SA-PINN
0.013
0.078
0.256 81
PirateNet
0.00591
0.05145
0.16819
HB-PINN
0.00071
0.01122
0.05865
sPINN
0.125 41
0.213 97
0.753 06
hPINN
0.068 73
0.156 08
0.557 50
MFN-PINN
0.139 45
0.232 43
0.794 08
1000
XPINN
0.102
0.196 43
0.680 79
SA-PINN
0.058 81
0.158 42
0.515 69
PirateNet
0.02666
0.11171
0.34722
HB-PINN
0.00364
0.02857
0.12837
sPINN
0.133 26
0.226 16
0.755 87
hPINN
0.080
0.17093
0.586 63
MFN-PINN
0.152 02
0.242 13
0.807 34
2000
XPINN
0.131
0.223
0.751 03
SA-PINN
0.075 84
0.176 81
0.570 22
PirateNet
0.04191
0.14184
0.42391
HB-PIN HB-PINN
0.02075
0.06640
0.29832
014
NERL PINN
012
J0PMN
Rirah shant
010
0.08
0.04
000
8.00
Reynolds Number
Figure 36: MSE Comparison with Reynolds Numbers
33

---

H
The more complex obstructed cavity flow model
H
Figure 37: Extended schematic of the segmented inlet configuration in an obstructed rectangular cavity flow.
To validate the capability of our Hybrid Boundary Physics-Informed Neural Network (HB-PINN) in handling geometrically complex scenarios, we extend Case from the main text to a more intricate configuration. As illustrated in 37 the computational domain is expanded to rectangular cavity of width 2 and height 1. Two types staggered obstructions are embedded within the cavity:
. Type A Obstructions:
Width 0.2, Height 0.5
Type Obstructions:
Width 0.2, Height = 0.25
The segmented inlet configuration (see Appendix E. for geometric specifics) imposes normal inflow velocity u 0.5, while the outlet, positioned at the upper-right corner of the cavity, enforces static pressure 0. All other boundaries adhere to no-slip conditions (u 0).
SITVI
L303
02
1.05
18 PBhinen
PMN
1233
1,05
4.00
07-
625
350
125
128
62t
679
335
135
1.55
GEOYSTI
Figure 38: Velocity predictions and residuals of sPINN versus HB-PINN in the extended obstructed cavity flow model. Left: Velocity predictions; Right: Absolute errors relative to CFD; Bottom: CFD-computed results.
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
34

---

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
I
Heat equation
To demonstrate the applicability of the HB-PINN method to other PDE systems, we evaluate its performance two-dimensional transient heat conduction problem, governed by the following equation:
aT
= (
82T
Öt 0r2
+
dy²
(12)
T(x, y, t) represents the temperature field, where the thermal diffusivity a = f. is defined by the thermal conductivity k, density and specific heat capacity The geometric configuration, shown in 3ig features square domain with edge length 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 dimensionless temperature 1, while external boundaries are fixed at O.
pe
Figure 39: Geometric configurations employed in heat conduction problems
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 two-dimensional square cavity, where the top lid is assigned horizontal velocity of u while the other walls maintain no-slip condition. The boundary conditions are mathematically formulated as follows:
1, 0 0< y=
0, others:
V = O, on 8Ω;
p = 0, (0.0).
The velocity predictions of sPINN and HB-PINN are shown in Fig 42 Due to the simpler geometric configuration of the model compared the cases discussed the main text, the improvement in accuracy is less pronounced than that observed in complex-boundary models. However, evidenced by residual results 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 boundary regions across diverse models. More detailed error metrics are summarized in the "LDC" section of Table
35

---

T20.25
FOG.S TS0.75
Ib1
Figure 40: Comparison of temperature results at 0.25, 0.5, 0.75, and 1.0 for the heat conduction problem: (a) sPINN predictions; (b) HB-PINN predictions; (c) ground truth (GT).
51
Figure 41: Residuals between predicted temperature and ground truth (GT) at 0.25, 0.5, 0.75, and 1.0 for the heat conduction problem: (a) Residuals of sPINN predictions:(b) Residuals of HB-PINN predictions.
SPINN
(B-PPINN(our)
CFD(GT)
1.00
0.75
0.50
0.25
0.00
Figure 42: Comparison of Velocity Predictions Between sPINN, HB-PINN, and CFD in Lid-Driven
Cavity Flow.
36

---

SPINN
HB-PINN(ours)
0.25
0.20
0.15
0.10
0.05
0.00
Figure 43: Residuals of Velocity Predictions Between sPINN, HB-PINN, and CFD Lid-Driven Cavity Flow
Table 6: Comparative analysis of error metries across the more complex obstructed cavity flow, heat equation, and Lid-Driven Cavity (LDC) flow
Error Metrics
Problem
Method
MSE())
MAE(()
Relative L2(4)
sPINN
0.215 89
0.257 29
0.838 66
Complex Cavity Flow
HB-PINN
0.01331
0.05637
0.20829
sPINN 0.002 76
0.017 80
0.107 73
Heat equation (1 1)
HB-PINN
0.00090
0.01071
0.06152
sPINN
0.000 0.00002
0.011 52
0.058
LDC
HB-PINN
0.00377
0.02001
K
Training and Inference Times
Table 7: Comparison of Training and Inference Times
Method
Training Time (1000 epochs)
Inference Time
SPINN
31.1
hPINN
4
MEN-PINN
92.5
XPINN
101.5
SA-PINN
41.4s
PirateNet
106.9s
HB-PINN
47.8s
0.29
CFD
17
37


Reviews:
[{"review_id": "swfuV2NBLW", "reviewer": "Reviewer_tZvQ", "summary": "The 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.", "questions": "- 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**", "limitations": "The 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.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "I do not want to change the score since the improvements are only incremental to further enhance the existing paper. The core contribution had already been mentioned in the paper earlier.", "rebuttal": "**For Q1**\n\n**A1**:We thank the reviewer for their valuable suggestions.\n\nTransient, vortex-dominated flow problems, such as the Kármán vortex street, are indeed classic and important models in fluid mechanics. However, solving these problems within a fully unsupervised learning framework remains a significant challenge for the PINN-based method due to the complex, non-linear interactions between vortices and turbulence. Since our focus is on solving the Navier-Stokes equations with complex boundaries in an unsupervised manner, while the primary challenge in vortex-dominated flow problems is accurately capturing transient turbulent characteristics, these cases are not included in the current version of our paper.\nIn response to the request, we tested our method on a transient Kármán vortex street model at $Re ≈ 250$ over $0.1s$ (encompassing two vortex shedding cycles) for further clarification. Both unsupervised and supervised learning settings, using 15 discrete data points within the conical wake region behind the cylinder, were tested.\n\nThe MSE calculation results are shown in the table1 below, where the error under unsupervised case is significantly larger compared to the supervised case. Although the MSE in the unsupervised case is still acceptable, it is important to note that, based on our observations, the primary error is concentrated in a small portion of the wake region behind the cylinder, which makes it difficult to accurately capture the Kármán vortex street flow. Although supervised wake predictions are relatively accurate within a few cycles, an accumulation of errors over time remains observable. This underscores that transient problems in long-term sequences present a key area for future research.\nWe will consider adding this case study to the appendix for further reference.\n\n**Table 1:** Mean Squared Error (MSE) comparison at different time points\n\n$$\n\\\\begin{array}{lccccc}\n\\\\hline\n\\\\text{Method} & T\\\\!=\\\\!0.02 & T\\\\!=\\\\!0.04 & T\\\\!=\\\\!0.06 & T\\\\!=\\\\!0.08 & T\\\\!=\\\\!0.10 \\\\\\\\\n\\\\hline\n\\\\text{Supervised}   & 7.45 \\\\times 10^{-5} & 1.47 \\\\times 10^{-4} & 1.93 \\\\times 10^{-4} & 2.45 \\\\times 10^{-4} & 3.16 \\\\times 10^{-4} \\\\\\\\\n\\\\text{Unsupervised} & 1.53 \\\\times 10^{-3} & 2.12 \\\\times 10^{-3} & 2.76 \\\\times 10^{-3} & 3.20 \\\\times 10^{-3} & 3.55 \\\\times 10^{-3} \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\n**For Q2**\n\n**A2**:The radial basis function (RBF), Gaussian kernel function, and indicator function you mentioned are indeed potential alternative approaches for distance metric networks.\n\n**The Gaussian kernel function** is the most commonly used type in **RBF**. An array of Gaussian kernels (or other RBFs) placed along boundaries constructs a signed approximate distance from spatial point x to those boundaries ($s(x) ≈ d(x)$). This approach resembles the R-function method mentioned in our paper, but simultaneously, its construction difficulty and time cost increase with boundary complexity.In contrast, when training distance functions via neural networks, computational cost remains unchanged as long as the network architecture is fixed.\n\n**The indicator function** proves straightforward for precisely locating boundaries or identifying distinct physical subdomains, offering value for multi-region problems. However, its key limitation lies in providing only discrete \"in/out of boundary\" information, lacking a continuous measure of proximity. Consequently, we consider it less suitable for pure distance metrics and better suited as an auxiliary feature for identifying regional affiliations.\n\nRegarding the **power-law function** we adopted, it requires special clarification: The core function of aforementioned methods like radial basis function (RBF) is to compute the approximate distance from spatial coordinate points to spatiotemporal boundaries. The power-law function, however, serves as a post-processing operation applied after obtaining this approximate distance. When the power exponent exceeds 1, it amplifies weight values in non-zero distance regions, with the amplification effect intensifying proportionally to distance magnitude. Therefore, as a post-processing step, the power-law function does not conflict with the aforementioned distance calculation methods (e.g., RBF) but operates in a complementary and mutually reinforcing manner.\n\n\n**For Q3**\n\n**A3**:Thank you for your valuable suggestion. We will follow this suggestion and include sectional profiles of velocity and pressure, along with the temporal history for the transient cases, in the appendix to provide a more comprehensive understanding of the performance of different approaches, particularly in regions where the flow-field gradients are strong.\n\n**For Q4**\n\n**A4**: We thank the reviewer for highlighting this point and prompting a clearer comparison. The key distinction between our work and that of Sun et al. (2020) lies in the construction of the distance field, which directly relates to the method's capability to handle complex geometric boundaries:\n\nSun et al.'s work was applied to a vascular model with a relatively simple geometric structure, for which an exact analytical expression for the distance function exists. Consequently, they directly used the analytical solution as the distance field without requiring additional learning.\n\nHowever, as demonstrated in complex configurations like Case 2 and Case 3 – which are the focus of this paper – analytical distance functions cannot be directly obtained. Computing the distance for such boundaries typically requires computing distances to individual boundary segments and then taking the minimum. Crucially, this minimum operation is non-differentiable, preventing its integration with the main network for joint backpropagation. Therefore, we employ a dedicated neural network (the distance metric network), trained using the minimum distance as labels, to learn a high-precision approximation of the distance field for complex boundaries. This neural network-based fitting approach is universal and can effectively obtain the required (and differentiable) distance function regardless of whether the boundary geometry is simple or complex.\n\n\n**For Q5**\n\n**A5**: We fully recognize the importance of time-evolving (unsteady) problems in fluid mechanics. In fact, they are much more complex than steady problems due to the difficulty of learning non-linear interactions over time. Instead of directly addressing dynamic scenarios with complex boundaries, our goal is to first provide a solution for steady cases and simple transient cases with complex boundaries, which have not yet been adequately addressed in previous methods.\n\nThrough our testing of the Kármán vortex street model at $Re ≈ 250$ (as mentioned in A1) and the recorded error profiles (Table 1), we observed that the core difficulty lies in capturing the time-evolving process within the wake region. The HB-PINN method we proposed does not incorporate specific optimizations for this aspect, and therefore currently also struggles to effectively solve such problems. This is a common challenge faced by current unsupervised PINN methods and represents a promising direction for future research.\n\n**For Q6**\n\n**A6**: We fully agree that computational cost is a key metric for assessing the practical value of a method. Table 2 details the comparative training and inference time costs for Case 2 under unified experimental settings (training on an NVIDIA RTX 4090 GPU, CFD run on an Intel Xeon E5-2640 v4 CPU). All PINN methods used the same 20,000 randomly sampled PDE collocation points.\n\n**Training Efficiency**:\nHB-PINN requires only marginally longer training time than the most fundamental sPINN and shows no disadvantage compared to other PINN variants. While maintaining low training costs, HB-PINN achieves significantly lower prediction errors, accomplishing an optimized balance between accuracy and efficiency.\n\n**Inference Efficiency**:\nWhen predicting results on a $200*200$ grid, the trained PINN method achieves inference times on the millisecond scale, whereas the CFD method requires 17 seconds. This orders-of-magnitude advantage endows HB-PINN with greater potential for real-time simulation, parameter scanning, and optimal control – scenarios demanding high-frequency access to flow field data.\n\n**Table 2:** Comparison of Training and Inference Times\n\n$$\n\\\\begin{array}{lcc}\n\\\\hline\nMethod & Training\\\\, Time\\\\, (1000\\\\, epochs) & Inference\\\\, Time \\\\\\\\\n\\\\hline\n\\\\text{SPINN}       & 31.1\\\\,\\\\text{s} & \\\\ 0.075\\\\,\\\\text{s} & \\\\\\\\\n\\\\text{hPINN}       & 48.3\\\\,\\\\text{s} & \\\\ 0.052\\\\,\\\\text{s} & \\\\\\\\\n\\\\text{MFN-PINN}    & 92.5\\\\,\\\\text{s} & \\\\ 0.051\\\\,\\\\text{s} & \\\\\\\\\n\\\\text{XPINN}       & 101.5\\\\,\\\\text{s} & \\\\ 0.029\\\\,\\\\text{s} & \\\\\\\\\n\\\\text{SA-PINN}     & 41.4\\\\,\\\\text{s} & \\\\ 0.034\\\\,\\\\text{s} & \\\\\\\\\n\\\\text{PirateNet}   & 106.9\\\\,\\\\text{s} & \\\\ 0.027\\\\,\\\\text{s} & \\\\\\\\\n\\\\hline\n\\\\text{HB-PINN(Ours)}     & \\\\mathbf{47.8\\\\,\\\\text{s}} & \\\\mathbf{0.042\\\\,\\\\text{s}} \\\\\\\\\n\\\\hline\n\\\\text{CFD}         & \\\\text{--} & 17\\\\,\\\\text{s} \\\\\\\\\n\\\\end{array}\n$$"}, {"review_id": "1NL4jM3A3D", "reviewer": "Reviewer_BSqT", "summary": "This 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.", "questions": "1.  **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.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**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.", "was_revised": true, "final_justification": "The author addressed most of my concerns, so I have raised my rating. As I am not an expert in this specific field, considering the overall quality of this paper, I still consider it to be borderline. I hope the authors can improve the overall quality of the figures and writing in the future.", "rebuttal": "**For Q1**\n\n**A1**: The essential innovation of HB-PINN is not simply employing neural networks to fit the $P$ and $D$ functions, but rather in systematically addressing the fundamental limitations of traditional hard-constraint methods in complex boundary scenarios. The conceptual breakthroughs are:\n\n**1. Breaking the Geometric Adaptability Bottleneck of Traditional Hard Constraints**\n\nExisting hard-constraint methods naively construct $P$ (only guaranteeing boundary point conditions) and $D$ (merely providing geometric distance), decoupling the boundary function $P$ from the solution satisfying the internal physics. While effective for simple boundaries (e.g., regular geometries, low-dimensional spaces), these methods fail to generalize to complex boundaries. This is because existing hard constraints construct $P$ solely considering boundary conditions, resulting in uncontrolled and often irregular output behavior within the spatial interior. This erratic behavior is highly detrimental to the training of the main network when handling complex boundaries.\n\n**2. Proposing a Dual-Network Synergistic Architecture for Complex Boundaries**\n\n**Differentiable Distance Metric Network ($\\\\mathcal{N}\\_{\\\\mathcal{D}}$)**: Provides a distance function. Used in a power-law form ($\\alpha$) to adjust the weight controlled by the distance function within the interior, ensuring the physics (governed by the PDE) dominates away from boundaries.\n\n**Boundary Feature Network ($\\\\mathcal{N}\\_{\\\\mathcal{P}}$)**: Models the physical feature distribution in the neighborhood of the boundary, rather than simply fitting boundary values. This ensures the output of $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ is controlled and well-behaved. Furthermore, by incorporating weak constraints derived from the PDE into its construction, $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ helps reduce the initial training difficulty for the main network when learning the PDE.\n\nOur ablation study (Section C.1 in the Appendix) separately compares the impact of the $P$ and $D$ functions on the results, demonstrating that both the proposed $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ and $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ networks play a positive role in solving complex boundary problems. Furthermore, existing research using hard constraints to solve physical models largely remains confined to simple geometric boundaries, with complex boundary problems largely unexplored. HB-PINN was developed precisely to address this unsolved challenge. Comparative evaluations against multiple PINN variants also demonstrate that HB-PINN achieves higher accuracy when handling complex boundary problems.\n\n**For Q2**\n\n**A2**: We understand the concern regarding the complexity of our architecture. However, the HB-PINN method fundamentally consists of three core subnetworks: the main solution network, the boundary condition network, and the distance function network. In our implementation, we define only these three subnetwork frameworks. Under each subnetwork, structurally identical neural networks are used to independently output each physical quantity. This approach is operationally more intuitive for applying constraints to different physical quantities and contributes to maintaining code simplicity and clarity.\n\nInitially, we experimented with a **Multi-output Network** (using a single neural network to simultaneously output all physical quantities to be solved). However, since it uses the same parameter space for computing each physical quantity, differing only at the output layer, this led to a significant reduction in prediction accuracy (see the \"Multi-output\" section in Table 1).Therefore, we considered decoupling the different physical quantities and employing **Specialized Networks** (each network outputs only one target physical quantity). This setup ensures that the computation of each physical quantity is unaffected by the parameter space of the others. Furthermore, training the partial differential equation inherently requires separately calculating the partial derivatives of different physical quantities. Consequently, this approach actually reduces the operational difficulty during the training process. All PINN methods in our paper use Specialized Networks to ensure the fairness of comparisons.\n\nMoreover, as mentioned in Q1 above, current hard-constraint methods struggle to address the Navier-Stokes equations with complex boundaries (please refer to the results shown in Figures 7 and 9 of our paper), even though they are not necessarily simpler than our method.\n\nThe reviewer's suggestion to merge the functionalities of multiple networks is a valid approach for structural optimization. However, integrating them within the hard-constraint architecture remains a significant challenge, representing a classic multi-task optimization problem. Solving this involves difficulties such as: task conflict, compounding and amplification of problem complexity, and challenges in resource allocation. In the context of this work, this manifests because the distance field network $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ and the boundary solution network $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ have fundamentally different learning objectives: $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ needs to accurately fit the geometric topology, while $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ needs to model the physical characteristics of the boundary layer. We will continue research in this direction, exploring ways to integrate different functionalities to optimize the network structure.\n\n\n\n\n**Table 1:** Performance comparison of network architectures\n\n$$\n\\\\begin{array}{l l c c}\n\\\\hline\n\\\\text{Method} & \\\\text{Network Structure} & \\\\text{MSE in Case 2} & \\\\text{Training time per 1000 epochs (s)} \\\\\\\\\n\\\\hline\n\\\\text{Multi-output Network} & [6\\\\times384]+3 & 0.0035 & 138.1 \\\\\\\\\n\\\\text{Specialized Networks} & 3\\\\times([6\\\\times128]+1) & 0.0008 & 47.8 \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\n**Table 2:** Training time comparison of different PINN methods (units: seconds)\n\n$$\n\\\\begin{array}{lccccccc}\n\\\\hline\n\\\\text{Method} & \\\\text{SPINN} & \\\\text{hPINN} & \\\\text{MFN-PINN} & \\\\text{XPINN} & \\\\text{SA-PINN} & \\\\text{PirateNet} & \\\\text{HB-PINN} \\\\\\\\\n\\\\hline\n\\\\text{Training time} & 31.1 & 48.3 & 92.5 & 101.5 & 41.4 & 106.9 & 47.8 \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\n**For Q3**\n\n**A3**: We thank the reviewer for the in-depth discussion of the boundary handling mechanism. We fully understand the theoretical simplicity advantage of the standard formulation $(q = BC(x) + D(x) * H(x))$ you proposed. However, HB-PINN introduces the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network—pre-trained with weak equation constraints instead of a simple $BC(x)$ function—to fundamentally address the challenges inherent in complex boundary scenarios.\n\nConstructing $BC(x)$ directly is feasible when the boundary geometry is regular and the physical distribution is smooth. However, in complex boundary scenarios involving features like multiple obstructions, multiple inlets, or sharp corners, directly constructing a suitable $BC(x)$ becomes extremely difficult or impractical.\n\nFor complex boundaries, training the solution network $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ solely to satisfy boundary conditions leads to distorted or discontinuous outputs within the spatial interior, particularly near junctions between different boundary types. This significantly increases the difficulty of training the main network ($\\\\mathcal{N}\\_{\\\\mathcal{H}}$), often to the point of preventing convergence.\n\nSpecifically, The significant difference between flow field inlet velocity conditions and wall no-slip conditions means that for complex boundaries, forcing the network ($\\\\mathcal{N}\\_{\\\\mathcal{P}}$) to satisfy both constraints simultaneously can result in uncontrolled behavior in its internal outputs.\n\nAs documented in our hPINN experiments within the paper. The main network loss could effectively decrease only after relaxing strict constraints on specific boundary regions (e.g., the wall regions near the inlet:$0 < x < 0.4$, $y=0$ \\& $y=1$). Without this relaxation, the main network loss started extremely high and failed to converge, making it impossible to obtain meaningful comparison results. However, this relaxation compromises the satisfaction of the full boundary conditions, which is unacceptable for solving physical models. This precisely illustrates the dilemma of traditional hard-constraint methods with complex boundaries.\n\nThe loss progression recorded in Table 3 clearly shows the original hPINN starting with an extremely high loss that failed to decrease normally, while the relaxed hPINN version started with a much more manageable loss value. This is a direct consequence of the uncontrolled $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ output under complex boundary conditions.\n\nCrucially, by incorporating hard constraints derived from the PDE into $\\\\mathcal{N}\\_{\\\\mathcal{P}}$'s training (see HB-PINN loss components in Table 3): The initial training loss is significantly reduced and the loss decreases normally during subsequent training.\n\nThis demonstrates that incorporating weak PDE constraints into the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network training effectively mitigates irregular behavior in its outputs within the interior domain. Furthermore, because $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ has undergone preliminary training guided by the governing equation, it substantially reduces the initial training difficulty for the main network ($\\\\mathcal{N}\\_{\\\\mathcal{H}}$). Therefore, adding weak PDE constraints to the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network training is fundamental and necessary.\n\n\n**Table 3:** Loss comparison across different methods at various epochs\n\n$$\n\\\\begin{array}{c c c c}\n\\\\hline\n\\\\text{Epoch} & \\\\text{Original hPINN} & \\\\text{Relaxed hPINN} & \\\\text{HB-PINN} \\\\\\\\\n\\\\hline\n0     & 2335810 & 47.44 & 7.129 \\\\\\\\\n5000  & 3318232 & 26.81 & 1.244 \\\\\\\\\n10000  & 5633070 & 25.59 & 1.070 \\\\\\\\\n15000  & 1076081 & 11.20 & 0.810 \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$"}, {"review_id": "I88A0WX2Nq", "reviewer": "Reviewer_Zovt", "summary": "A 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.", "questions": "Points **(2)**, **(3)**, and **(5)** in the \"Strengths And Weaknesses\" section  are my questions particularly affecting the evaluation.", "limitations": "Yes.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 2, "strengths_and_weaknesses": "### 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.", "was_revised": true, "final_justification": "The response mostly addressed my concerns, and thus I raised the rating. With that being said I cannot assess how the paper can be improved to make the contribution clearer (as done in the response), not only to me but to the general audience, thus my rating remains around borderline.", "rebuttal": "**For Q1&2(W1&2)**\n\n**A1&2**: We thank the reviewer for their valuable comments. These questions help us clarify the motivation behind our method's design and its differences from hPINN more clearly. We fully agree with the reviewer's point regarding the importance of detailing our design choices and their underlying rationale.\n\n**1. Why use $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ instead of the original boundary condition function?**\n\nConstraining the solution network $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ solely with boundary conditions works when the boundary structure is simple. However, for complex boundaries, this approach leads to distorted or discontinuous outputs within the solution domain, especially near junctions between different boundary types. 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 (e.g., wall regions near the inlet, more details described in Q4) had to be relaxed for the main network loss to decrease effectively. Otherwise, the loss became extremely large and failed to converge, making it impossible to obtain valid comparison results (as shown by the initial loss descent trend in Table 1). \n\nHowever, 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 $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ as an alternative solution aimed at more robustly handling complex boundary constraints.\n\n**2. What is the rationale for the necessity of $\\\\mathcal{N}\\_{\\\\mathcal{D}}$? What is the significance of training the $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ network when an analytical distance $\\\\hat{\\\\mathcal{D}}\\_{q}$ can be directly computed? Why not use the analytical $\\\\hat{\\\\mathcal{D}}\\_{q}$ directly in Equation (3)?**\n\nFor geometrically simple models (like the vascular model mentioned in Sun et al. (2020) [11]), analytical expressions for the distance can indeed be easily obtained. However, for most complex geometries, computing the distance typically requires calculating distances to individual boundary segments and then taking the minimum. Crucially, this minimum operation is non-differentiable, preventing its integration with the main network for joint backpropagation. Even if approximate expressions like the R-function can be used, constructing the corresponding approximate function representation for complex geometries remains highly complex (we believe its construction difficulty is even greater than training a distance function using the $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ network). More importantly, compared to using the distance network, analytical/approximate methods significantly increase the time cost of training the main network (approximately 2:1). This is because the neural network's inherent auto-differentiation capability enables seamless integration with the main PINN network, ensuring efficiency during joint training and gradient backpropagation. Based on these considerations, we chose the more convenient and efficient $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ network to construct the distance function. This is a design that balances universality and efficiency for handling complex geometries.\n\n**Table 1:** Loss comparison across different methods at various epochs\n\n$$\n\\\\begin{array}{c c c c}\n\\\\hline\n\\\\text{Epoch} & \\\\text{Original hPINN} & \\\\text{Relaxed hPINN} & \\\\text{HB-PINN} \\\\\\\\\n\\\\hline\n0     & 2335810 & 47.44 & 7.129 \\\\\\\\\n5000  & 3318232 & 26.81 & 1.244 \\\\\\\\\n10000  & 5633070 & 25.59 & 1.070 \\\\\\\\\n15000  & 1076081 & 11.20 & 0.810 \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\n**For Q3(W3)**\n\n**A3: Regarding the Motivation for $\\\\lambda\\_1 > 0$:**\n\nThe core motivation for setting $\\\\lambda\\_1$ to a non-zero value (albeit much smaller than $\\\\lambda\\_2$ and $\\\\lambda\\_3$) in the loss function of the particular solution network $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ is to introduce weak constraints derived from the PDE. This design aims to:\n\n**1.Mitigate Internal Irregularity:** Constraining $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ solely with boundary conditions (i.e., $\\\\lambda\\_1=0$) often leads to uncontrolled (irregular or distorted) solutions within the solution domain interior, particularly away from boundaries.\n\n**2.Reduce Main Network Training Difficulty:** Imposing preliminary weak PDE constraints during training significantly increases the controllability of the solution produced by the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network. This results in an initial solution from $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ that is closer to the true physical field, providing a better starting point for the subsequent training of the main network. This effectively reduces the initial training difficulty for the main network (this is closely related to the dilemma of handling complex boundaries with boundary constraints alone, as revealed in our responses to A1&2). The beneficial effect of $\\\\lambda\\_1 > 0$ on initial loss is reflected in the HB-PINN loss components documented in Table 1.\n\n**Regarding the Ablation Study:**\n\nWe fully agree with the reviewer on the importance of ablation studies for validation and have presented detailed results in Appendix C.1. Specifically:\n\n1.The mD configuration in the experiment corresponds to the case where $\\\\lambda\\_1 = 0$.\n\n2.The error results in Appendix Table 2 clearly show that setting $\\\\lambda\\_1 > 0$ reduces the solution error by approximately an order of magnitude compared to $\\\\lambda\\_1 = 0$ (mD). It is crucial to emphasize that this significant advantage was achieved under conditions where we appropriately relaxed constraints on certain boundary segments (relevant details will be elaborated in Q4).\n\n3.Furthermore, the loss records from the experiments provide additional evidence: $\\\\lambda\\_1 = 0$ results in a poor-quality solution from the trained $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network, which is severely detrimental to the subsequent training of the main network (specific data can be referenced in the original hPINN loss records in Table1).\n\n**For Q4(W4)**\n\n**A4**: We thank the reviewer for their attention to the rigor of experimental details and for highlighting the importance of clearly explaining the boundary condition relaxation operation.\n\nIn the hPINN comparative experiments in this paper (including the mD configuration in ablation study C.1), we selectively relaxed the strict constraints on the wall boundaries near the inlet **(Due to the uncontrolled solutions generated within the solution domain, as mentioned in A1,2, and A3)**. \n\n To enable the hPINN method to produce a trainable solution, we proactively reduced the enforcement strength of the wall constraints in the conflict region near the inlet (specifically, $0 < x < 0.4, y=0$ \\& $y=1$ in Case 2). This is evident in the error plot for the mD configuration in Fig. 13 of Appendix C.1, where significant errors near the upper and lower walls close to the inlet are observable, precisely resulting from the relaxation of the relevant wall constraints.\n\nIt is crucial to reiterate: While this operation allows the main network loss to decrease and seemingly yields results closer to Ground Truth, it is fundamentally an expedient measure that violates physical reality.\n\nWe will add more detailed figures in the appendix illustrating the reasons for relaxing these constraints.\n\n**For Q5(W5)**\n\n**A5**: We thank the reviewer for raising this important comparison suggestion, which helps more comprehensively evaluate the contribution of the method's components. The improvement strategy you inquired about—namely, separately removing the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ and $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ modules—has been validated in the ablation study presented in Appendix C.1.\n\nIn Appendix C.1, we designed two configurations, mD and mP, for comparison:\n\n**1.mD Configuration:** This configuration used only the distance function network $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ and did not use the particular solution network $\\\\mathcal{N}\\_{\\\\mathcal{P}}$. For the same reason explained in the previous Q4 response (i.e., the intense physical conflict at the wall boundaries near the inlet), the mD configuration also required relaxing the boundary constraints in that region to ensure trainability.\n\n**2. mP Configuration:** This configuration used only the particular solution network $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ and did not use the distance function network $\\\\mathcal{N}\\_{\\\\mathcal{D}}$. The experimental results show that the solution errors for both the mD (only $\\\\mathcal{N}\\_{\\\\mathcal{D}}$) and mP (only $\\\\mathcal{N}\\_{\\\\mathcal{P}}$) configurations were significantly lower than those of the original hPINN method. Furthermore, the error of the complete HB-PINN model, which integrates both $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ and $\\\\mathcal{N}\\_{\\\\mathcal{D}}$, was further significantly reduced. This demonstrates that both our proposed $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ and $\\\\mathcal{N}\\_{\\\\mathcal{D}}$ networks played a positive role in improving the model's accuracy.\n\n\n\n**For Q6(W6)**\n\n**A6**: We thank the reviewer for their attention to the reproducibility of the experiments. All baseline methods maintain consistency in the following core parameters:\n\n**Table 2:** Uniform configuration for all baseline methods\n\n$$\n\\\\begin{array}{lc}\n\\\\hline\n\\\\text{Parameter Category} & \\\\text{Unified Configuration} \\\\\\\\\n\\\\hline\n\\\\text{Network Architecture} & 6\\\\ \\\\text{hidden layers} \\\\times 128\\\\ \\\\text{neurons} \\\\\\\\\n\\\\text{Training Iterations} & 3\\\\times10^{5} \\\\\\\\\n\\\\text{Optimizer} & \\\\text{Adam} \\\\\\\\\n\\\\text{Initial Learning Rate} & 1\\\\times10^{-3} \\\\\\\\\n\\\\text{Points per Epoch} & 20,000\\\\ \\\\text{(random sampling from domain)} \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\nWe will add more detailed configuration details for the other baseline methods in the appendix."}, {"review_id": "Oabc3FmI75", "reviewer": "Reviewer_qJeV", "summary": "The 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.", "questions": "1. 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?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\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).", "was_revised": true, "final_justification": "The response mostly addressed my concerns, and I raised the score. Some issues remain to be resolved in future work, and I understand that these are common challenges faced by PINN-type methods, which may be difficult to overcome in the short term. At the very least, this paper has made effective attempts and progress in this field.", "rebuttal": "**For Q1**\n\n**A1**: The performance of HB-PINN is indeed dependent on $\\\\alpha$, and either excessively large or small values of $\\\\alpha$ adversely affect training.\n\nIf $\\\\alpha$ is too small (e.g., $\\\\alpha=1$), the weight of the PDE loss in the interior domain becomes insufficient. Conversely, if $\\\\alpha$ is too large (e.g., $\\\\alpha=20$), the boundary layer becomes excessively thin. Both scenarios impede effective training of the main network. We conducted relevant ablation studies in Appendix C.3. Generally, when $\\\\alpha$ falls within the range of 5–10, the final training results of the main network vary minimally. Therefore, selecting $\\\\alpha$ within the 5–10 range is optimal.\n\nBased on observations and repeated experiments across multiple different models, we can provide the following recommendations for the selection ranges of the key parameters $\\\\alpha$ and the number of pre-training iterations for the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network:\n\n1.The distance metric network parameter $\\\\alpha$ can be chosen between 5 and 10, adjusted according to the density of obstacles (i.e., the distance between different boundaries) within the domain. Sparser obstacle distributions correspond to smaller $\\\\alpha$  values.\n\n2.The number of pre-training iterations for the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network can be chosen between 10,000 and 50,000, adjusted based on the complexity of the boundary conditions. Higher complexity warrants selecting a higher number of iterations to ensure strict satisfaction of the boundary conditions.\n\nHowever, we have not yet been able to summarize an adaptive adjustment strategy for $\\\\alpha$ and the $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ pre-training iteration count that suits all scenarios. In future work, we will continue research on hyperparameter tuning to develop adaptive parameter selection schemes applicable to different situations.\n\n\n**For Q2**\n\n**A2**: We thank the reviewer for their interest in the computational efficiency characteristics of HB-PINN. Table 1 details the comparative training and inference time costs for different PINN methods on Case 2 under unified experimental settings (training on an NVIDIA RTX 4090 GPU, CFD run on an Intel Xeon E5-2640 v4 CPU). All models uniformly accept inputs with a shape of $[20000, 2]$.\n\n**Training Efficiency:**\n\nThe training time for HB-PINN (47.8 s) is significantly better than most advanced PINN variants (MFN-PINN: 92.5 s, XPINN: 101.5 s, PirateNet: 106.9 s). It is comparable to hPINN (48.3 s) and SA-PINN (41.4 s), and only slightly longer than the most basic sPINN (31.1 s). The training speed of certain methods (such as MFN and PirateNet) is reduced due to complex mathematical operations beyond the model architecture itself. Critically, while maintaining training costs on par with advanced methods, HB-PINN achieved significantly lower prediction errors, achieving an optimized balance between accuracy and efficiency.\n\n**Table 1:** Performance comparison of different PINN methods\n\n$$\n\\\\begin{array}{l c c c c c c c}\n\\\\hline\n & \\\\text{SPINN} & \\\\text{hPINN} & \\\\text{MFN-PINN} & \\\\text{XPINN} & \\\\text{SA-PINN} & \\\\text{Pirate Net} & \\\\text{HB-PINN} \\\\\\\\\n\\\\hline\n\\\\text{Training time (s)} & 31.1 & 48.3 & 92.5 & 101.5 & 41.4 & 106.9 & 47.8 \\\\\\\\\n\\\\text{Params (M)} & 0.30 & 0.34 & 0.36 & 0.16 & 0.30 & 0.06 & 0.36 \\\\\\\\\n\\\\text{FLOPs (G)} & 5.92 & 6.65 & 7.03 & 1.49 & 5.92 & 1.73 & 7.11 \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\n**For Q3**\n\n**A3**: We thank the reviewer for raising this crucial question concerning the universality of the method. Solving three-dimensional (3D) transient flow models under more realistic conditions is both a necessary direction for PINN development and a formidable challenge. \n\nThe HB-PINN framework is inherently capable of extension to 3D. We have currently tested it on a relatively simple 3D model (3D steady-state cavity flow), achieving an error range of $10^{-3}$ to $10^{-4}$ across multiple calculated 2D cross-sections. However, due to time constraints, more detailed comparisons and testing on complex 3D transient cases have not yet been conducted. Solving more complex and realistic problems remains a focus for future research.\n\nTo support this goal, we have established a laboratory setup for generating flow fields to conduct complementary physics-based experiments. In future work, we plan to cross-validate our computational results with real experimental data to better address the challenges associated with three-dimensional flow problems.\n\n\n\n**For Q4**\n\n**A4**: We sincerely thank the reviewer for highlighting the importance of addressing moving boundaries and time-varying boundary conditions, which are indeed representative of many real-world scenarios and hold significant research value.\n\nAlthough our current work does not directly address these challenges, we believe that HB-PINN has inherent potential to handle moving or time-varying boundary conditions for the following reasons:\n\n1.The $\\\\mathcal{N}\\_{\\\\mathcal{P}}$ network incorporates weak constraints derived from the PDE. This enables it to provide a physically reasonable initialization for the main network’s training across diverse boundary conditions, reducing sensitivity to temporal variations.\n\n2.The main network ($\\\\mathcal{N}\\_{\\\\mathcal{H}}$) focuses solely on satisfying the governing equations. This design minimizes interference from loss terms related to time-varying boundaries.\n\nTherefore, we believe HB-PINN possesses inherent potential to handle moving or time-varying boundary conditions. Future work will explicitly investigate these challenging scenarios.\n\n\n\n\n**For Q5**\n\n**A5**: We appreciate the reviewer’s interest in the broader applicability of HB-PINN. As mentioned in Q3\\A3, we have established a dedicated laboratory setup capable of generating controlled flow fields at a $1\\\\,\\\\text{m} \\\\times 1\\\\,\\\\text{m} \\\\times 1\\\\,\\\\text{m}$ scale to simulate atmospheric phenomena. However, there are still technical challenges in accurately capturing flow velocity. We are actively working to address these issues and plan to evaluate our method under real-world experimental conditions in the near future. This will serve as an important first step toward extending HB-PINN to industrial-scale applications.\n\n\n\n\n\n**For Q6**\n\n**A6**: At the current stage of our research, we primarily use CFD simulation results as the benchmark (Ground Truth) to evaluate the accuracy of HB-PINN compared to other PINN methods. This approach is widely adopted in PINN research due to the technical challenges associated with directly validating physics-informed neural networks against real experimental data.\n\nTherefore, we are currently only able to compare efficiency against CFD solvers. When predicting results on a $200*200$ grid, the trained PINN method achieves inference times on the millisecond scale, whereas the CFD method requires 17 seconds(as Table2). This orders-of-magnitude advantage endows HB-PINN with greater potential for real-time simulation, parameter scanning, and optimal control – scenarios demanding high-frequency access to flow field data.\n\n**Table 2:** Inference time comparison of different methods (in seconds)\n\n$$\n\\\\begin{array}{l c c c c c c c c}\n\\\\hline\n & \\\\text{sPINN} & \\\\text{hPINN} & \\\\text{MFN-PINN} & \\\\text{XPINN} & \\\\text{SA-PINN} & \\\\text{Pirate Net} & \\\\text{HB-PINN} & \\\\text{CFD} \\\\\\\\\n\\\\hline\n\\\\text{Inference time} & 0.075 & 0.052 & 0.051 & 0.029 & 0.034 & 0.027 & 0.042 & 17.000 \\\\\\\\\n\\\\hline\n\\\\end{array}\n$$\n\n\nRegarding the accuracy comparison with CFD solvers, we are eager to conduct such tests once our laboratory setup is fully operational."}]

Author rebuttal:

Meta-review: The paper proposes a novel PINN framework with solid technical contributions and clear motivation. Reviewers were generally positive, highlighting the methodological soundness and empirical results, with only minor concerns about scope and comparisons. The rebuttal provided satisfactory clarifications that strengthened confidence in the work. With overall ratings leaning positive, I recommend acceptance.
Decision: Accept (poster)