Physics-informed machine learning with domain decomposition and global dynamics for three-dimensional intersecting flows
Authors: Leslie K Hwang
Venue: NeurIPS 2025 poster ()
Keywords: Physics-informed neural networks
domain decomposition
global dynamics
three-dimensional fluid dynamics
Primary area: machine_learning_for_sciences

Abstract: Physics-informed neural networks (PINNs) have emerged as a promising framework to develop complex scientific surrogate models, yet their scalability and accuracy often degrade in non-canonical geometries, such as non-rectangular domains or three-dimensional (3D) domains with high aspect ratios. These limitations hinder the broader adoption of vanilla PINNs in real-world, practical systems. In this work, we introduce a multi-domain PINN (MDPINN) framework designed to address the scalability and generalization challenges inherent in 3D non-rectangular domains governed by nonlinear fluid dynamics. The target domain consists of intersecting 3D fluid channels with a high aspect ratio, inducing complex flow features such as deflections, mixing, and recirculations. Our approach is grounded in two key innovations: 1) domain decomposition, which partitions the channel volumes into multiple cubic-like subdomains, each modeled by an individual PINN, 2) enforcement of global dynamics (MDPINN-GD), which ensures that the total mass flow rate entering the domain equals that exiting. These innovations reduce the complexity of the problem imposed on individual PINNs and guide effective network optimization toward physically consistent solutions throughout the domain. We demonstrate that our method achieves: 1) 74.8\% accuracy improvement over a single-network PINN, and 2) 52.9\% accuracy improvement over MDPINN that do not enforce global mass conservation. Furthermore, the MDPINN-GD framework exhibits accurate prediction even in highly complex regions-such as the channel intersecting zone and the outlet zone characterized by intense flow mixing and large velocity gradients-achieving maximum normalized mean absolute errors below 14.9\% for velocity predictions compared to simulation results. This work establishes a path towards scalable, physically grounded surrogate modeling approach that is extensible to multiphysics and high-dimensional scientific problems.

Physics-informet machine learning with domain decomposition and global dynamics for three-dimensional intersecting flows
Leslie K. Hwang
School of Electrical, Computer and Energy Engineering
Arizona State University
Tempe, AZ 85287
kkwanggasu edu
Abstract
Physics-informed neural networks (PINNs) have emerged as a promising frame- work to develop complex scientific surrogate models, yet their scalability and accuracy often degrade in non-canonical geometries, such non-rectangular do- mains ftrre-dimensional (3D) domains with high aspect ratios. These limitations hinder the broader adoption of vanilla PINNs in real- world, practical systems. In this work, we introduce a multi-domain PINN (MDPINN) framework designed address the scalability and generalization challenges inherent in 3D non-rectangular domains governed by nonlinear fluid dynamics. The target domain consists of intersecting 3D fluid channels with high aspect ratio. inducing complex flow features such as deflections, mixing, and recirculations. Our approach is grounded in two key innovations: 1) domain decomposition, which partitions the channel volumes into multiple cubic-like subdomains, each modeled by an individual PINN, 2) enforcement of global dynamics (MDPINN-GII -DD),, which ensures that the total mass flow rate entering the domain equals that exiting. These innovations reduce the complexity of the problem imposed individual PINNs and guide effective network optimization toward physically consistent solutions throughout the domain. We demonstrate that method achieves: 1) 74.8% accuracy improvement single-network PINN, and 2) 52.9% accuracy improvement over MDPINN that do not enforce global mass conservation. Furthermore, the MDPINN-CN G- framework exhibits accurate prediction even in highly complex regions-such as the channel intersecting zone and the outlet zone characterized by intense flow mixing and large velocity moddients-achieviie maximum normalized mean absolute errors below 14.9% for velocity predictions compared to simulation results. This work estab- lishes a path towards scalable, physically grounded surrogate modeling approach that is extensible to multiphysics and high-dimensional scientific problems.
Introduction
Recent advancements in scientific machine learning have garnered much attention for their potential bridge modern machine learning methods with traditional scientific computing. This integration is particularly appealing to the engineering design community, where conventional simulation methods such computational fluid dynamics (CFD) be computationally expensive for their applications. For instance, CFD-based design optimization of fluidic networks-such as microchannels used for electronic cooling-often involves simulations across wide range of geometric and physical parame- ters. Major bottlenecks in these simulations lie in the generation and refinement of computational meshes for large geometric variations, as well as in the computational cost of iterative design searches.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

To address the challenges in engineering designs, physies-informed neural networks (PINNs) have emerged as viable surrogate modeling approach 9
PINNS are a class of neural networks whose solutions are guided by embedded physical laws rather than being solely determined by data. Unlike traditional data-driven machine learning approaches, it is demonstrated that PINNs do not require large datasets-or any datasets in simple cases-as they are trained using a thysicc-driven loss function formulated from the residuals of governing partial differential equations (PDEs), boundary conditions, and initial conditions. In particular, in high- dimensional fluid simulations, PINNs offer significant potential to reduce computational costs, where traditional CFD methods often require substantial computational time and resource demand as the dimensionality of a problem increases. In contrast, PINNs enable efficient inference once trained and can further extend scalable acceleration through transfer learning. For example, Tang et al. used transfer learning to model vortex-induce 87.5% (using only 1/8 of the original data), while achieving 10-fold decrease in computation time with a maximum relative error of 8.51% [1O]. Prantikos et al. used transfer learning to accelcrate the prediction of nuclear reactor transients, achieving at least 16-fold acceleration in training convergence, with prediction times under sub 10 seconds [Α.. In another study, Ohashi et al. applied a transfer learning scheme to PINNs for modeling two-dimensional (2D) transient temperature fields, achieving a f.9-fold reduction in computational time-from 8.32 to 0,84 hours [2] Lastly, coupled with extreme learning machines (ELM), a thysics-informed ELM demonstrated training time reduction from eight hours to a few seconds for simple ID and 2D linear PDEs [13] These results highlight the potential of PINNs as nowward-predictio models that scale to complex, multi-scale geometries and adapt to diverse flow conditions. Consequently. PINNs are promising methodologies solve challenging high- dibbeiinnnional physics problems, such as modeling ill-posed fluid systems where conventional CFD methods struggle due to under-constrained conditions and the existence of multiple plausible solutions [1 [4). PINNs have been successfully applied variety of problems, including steady and unsteady heat conduction, convection, and incompressible Navier-Stokes problems, offering mesh-free and differentiable alternative to conventional numerical
vibrations, reducing the required training data by
solvers [5] 15
Despite the advantages of PINNs for fluid simulations, their application to ttree-dimensional (3D) fluid systems with complex flow structures remains challenging. Chan et al. systematically evaluated the performance of PINNs against supervised learning approach trained on extensive CFD data for steady and pulsatile flow in a 3D curved tube with varying geometries. Their findings showed that PINNs were only successful under carefully controlled steady-state conditions, including (1) the use of a transformed coordinate system, (2) enforcement of no-slip boundary conditions as hard constraints, and (3) specialized strategies to propagate boundary effects (16.. The complexity of 3D domains amplifies challenges already encountered in 2D simulations-nie most notably, the handling of corner singularities. In 2D, these singularities typically occur at domain corners where abrupt changes in boundary conditions or geometry lead to discontinuities or sharp gradients in flow variables. In 3D, the problem becomes more pronounced as these singularities extend along edges or lines formed by the intersection of surfaces [8]. Cai et al. explored transient 3D flow reconstruction around a cylinder at Reynolds number 200, which required a large number of simulation data (at least 344,650 data points) and collocation points (3x 10') for PINN training [5]. In another study, Moser et al. investigated PINNs applied flow through cylindrical and aneurysmal geometries, using up to 17x 10t collocation points and enforcing mass conservation via Monte Carlo integration across continuity planes 19. While Deep Galerkin Methods provided the best for simple geometries-the the models still exhibited significant pressure errors in complex aneurysms. Collectively, these studies highlight the current limitations of 3D PINNs, even with relatively simple geometries. Notably, existing work has largely focused geometries with nearly uniform aspect ratios, where the strcamwise and spanwise dimensions are roughly equal.
In most of the literature, PINNs based on single neural network have shown limited effectiveness in modeling high-aspect-ratio 3D fluid channels (0).. Our separate empirical studies find that this occurs around a channel length that exceeds the depth by factor greater than five- -pimmarlly due to challenges in convergence and solution accuracy. Empirical studies on various geometries can be found in Appendix A.I High-aspect-ratio domains introduce pronounced disparities in characteristic length scales, often resulting in vanishing gradients during training and significantly imbalanced loss contributions across physical equations along different orientations. As consequence, single- network PINNs tend to prioritize dominant flow directions and large-scale structures while underrepresenting
2

---

transverse or localized features critical to accurate prediction. This imbalance can lead to poor generalization, limited predictive accuracy, and unstable convergence behavior. To address these issues, recent studies have proposed decomposing the computational domain and assigning dedicated subnetworks to individual physical components (e.g., momentum, continuity, energy). Multi-network,, multi-domain PINN (MDPINN) architecture with domain decomposition strategies have demonstrated greater effectiveness for such geometrically complex systems. One such example is given by a study investigating 3D blood flow in three-branch intersection, in which the L2 error was around 7.8% but required 8 10 epochs 12.. Another is given by Shukla et al. in which 2D steady-state lid driven cavity flow and heat conduction were investigated (2). The authors showed advancements in computational time by leveraging parallel tasks. Next, Laubscher [23] and Merdasi et al. [24] demonstrated that multi-network trchitectures-wiie within subdomains or for specific governing equations-cal canion significantly improve training stability and accuracy. This method is further described by Dwivedi et al., which introduced distributed PINN for each subdomain with additional flux conditions at cell interfaces as local regularizer when combining together, and demonstrated higher accuracy and better data efficiency over PINNs (25). Other various interface conditions and irregular domain geometries have also been studied by Jagtap al. 26 (27). Due to many interface conditions, Li et al. proposed meta-learning strategy to dynamically determine the appropriate constraints (28) However, these studies explored domains limited to 2D with simple constraints. These approaches mitigate the complexity faced by any single network, making them particularly suitable for high-aspect-ratte geometries characterized by anisotropic behavior and multiscale physics. Advancements, such as causal sweeping strategies, loss term balancing, and transfer learning, have improved training time and accuracy to certain degree, yet more research is required in higher-dimensional domain [29 20 30
subnetworks are optimized independently
Here, we investigate a MDPINN strategy and further improve with global dynamic enforce- ment, applied to 3D intersecting fluid channels characterized by large aspect ratios of 7.1:1.4:1 ddhph))) and strong flow mixing at the intersection. Fluid channel networks that promote flow mixing have broad applications, particularly in cold plates for electronics cooling, where flow mixing disrupts thermal boundary layer development and improves localized heat removal. Another application is in species mixing (e.g., chemical, particulates, fuel/air), in which homogeneous mixture is desired. The overall model consists of an ensemble of PINNs optimized jointly with parallel parameter updates. To ensure physical consistency across subdomains, we introduce two additional loss components: (1) global continuity loss (MDPINN-GD) and (2) matching boundary condition losses, which enforce conservation laws and interface compatibility, respectively.
2 Problem setup
2.1 Design domain
Cross-flow channels are widely used in fluidic networks due to their simple design and ability to create localized mixing zones that enhance fluid homogeneity and further heat transfer efficiency. Fig. (a) illustrates a 3D cross- cross-shaped channel configuration, where two inlet ports are positioned on the left and bottom faces, two outlet ports the right and top faces and two channels intersect in the middle. This configuration establishes a central cross- flow region that promotes fluid interaction and recirculation. The channel dimensions depth d. Liquid water at 25 O, with a kinematic viscosity of v = 8.917 x 10-7 m²/s and density of Po 999 kg/m², enters both inlets uniform velocity (Un) of 2.68 m/s. The objective is unsupervised learning, predicting the velocity and pressure fields within the domain using governing equations and boundary conditions only, without any simulation data experimental measurements. The flow regime is characterized by the Reynolds number [Re = DO /v]. where DH is the hydraulic diameter. By tuning Re, various flow regimes can be explored, but limit the scope of this work, Re is fixed to 247.5, 55, with system-wide aspect ratios of 7-=l/w /w = 5 and 72=1/d= = 7.14. To evaluate the accuracy of the PINN predictions, we use reference CFD simulation conducted in Ansys Fluent. The CFD domain is discretized at resolution of 1 pm, resulting in approximately 6.3 x 10 cells. The model assumes laminar flow with normalized convergence criteria of xx 10 for all residuals.
set to 500 um in length I, 100 μm in width w, and 70 um in
3

---

Muttiple subdomains
Outlet
(a)
(b)
1
Outlet
Inlet
Length
Outlet
Matching (1,2)'s interface
Matching (8,9)'s interface
Inputs:
Inputs:
x
Width
Multiple parallel networks
Inlet
Depth
Inlet
Update all subnetworks simultaneously
x-velacity
x-velocity
y-velocity
Outputs:
y-velocity
Loss: Summation across
z-velocity
Inter-network Losses
z-velocity
subnetwork losses
Outputs:
pressure
pressure
N
by
Ly
Ln²
Loc
Lod
Loc
1-1
Figure 1: (a) 3D cross-intersecting fluid channel. (b) MDPINN network architecture in which each subdomain is assigned a separate PINN. Network loss includes local and interface terms. Index i runs from through 9, corresponding to the nine subdomains. Index runs from 1 through 8, corresponding to the eight subdomains interfaces.
2.2 Governing equations
For demonstration purposes, we focus on a simplified flow regime, steady-state laminar flow, gov- erned by the 3D steady-state incompressible Navier-Stokes equations. improve generalization and numerical stability, the governing PDEs are non-dimensionalized. allowing the system to be charac- terized in terms of dimensionless parameters. Furthermore, input and output variables are normalized to the range of 0 to 1, which align with the effective range of common activation functions used in PINNs (e.g., tanh and swish (2). Consequently, the problem formulation incorporates the following normalized terms, denoted by to signify the dimensional parameters with x xx" /l, v" /w, = /d, = U* /Uo. = VV/UU. W = W /Uo, and = P"/PoU x, y, z are dimensionless coordinates in length, width, and depth directions, respectively, defined by the dimensional variables x² E [0, y" E [0, w]. and =* E |0, d]. U, V, W are the dimensionless velocity components the rr. V-, and z-directions, respectively, normalized Ug- P is the dimensionless pressure, normalized by PoU?. The non-dimen con-dimensional PDEs of continuity, I-, y-, and z-momentum in Cartesian coordinates are listed in order at Eq.
0U
OV
OW
dr 0U
+
71
Oy
72
Oz
OU
OU
OP
1
PU
OU
OU
dr
+
V
+ W
W
Oz
dr
Re
Ox
11
Oyy
12
022
0
(1)
0V
@V
0V
0P
1
08V
8V
0V
+ m1 V
Oy
+ W
Oz
71
Oy
Re
0r2
71
0y2
12
022
OW
OW
OW
0P
1
[@WW
PW
8
Or + +
+ 7/2
12
1
72
Oy
Oz
Re
Oz2
0y2
0=2
3 Multiple domain physics-informed neural network (MDPINN)
To address the optimization challenges associated with training single-network PINN on 3D domains with high aspect ratios, we adopt domain decomposition strategy that improves both convergence and scalability. As shown by the black dotted lines in Fig. (a), the domain is divided into nine subdomains with reduced aspect ratios. Each subdomain measures 100 um in length, 100 um in width, and 70 μm depth, resulting in nearly cube-like volumes with an aspect ratio 1.4:1.4:1. An overview of the MDPINN framework is shown in Fig. (b), where each subdomain is assigned to

---

individual subnetwork. Domain decomposition allows each subnetwork to learn within a subdomain with reduced geometric aspect ratios, which helps mitigate vanishing gradient issues and imbalanced loss contributions commonly observed in single-network PINNs applied to elongated geometries. The training loop shows both intra-network losses (within each subnetwork) and inter-network losses (across subnetworks). The losses are aggregated across all subnetworks into total loss function, which minimized through a single backpropagation step. This scheme enables simultancous parameter updates across all subnetworks.
Each subnetwork implemented as fully-connected network that takes 3D Cartesian coordinates as input and predicts four flow variables as output: P. U, V, and W. Based on literature and empirical hyperparameter tuning, each subnetwork and training settings are configured as follows with the range explored in parentheses, offering balance between accuracy and computational efficiency: 1) hidden layers (5-10), 2) 100 neurons per layer (100~150), 3) 100,000 epochs for Adam optimizer (50K~100K), 4) 20,000 collocation points (10K-40K) randomly sampled within each subdomain using Latin hypercube sampling [3I], and 5) boundary points uniformly sampled from grid with μm spacing (I jum) The activation function used is the adaptive sine [32]. Network optimization was conducted in two stages-the Adam optimizer with a learning rate of 1x10%, followed by the L-BFGS optimizer until convergence. The network was built using PyTorcl and its associated libraries. pseudocode outlining the network training process is included in Appendix
A.22 A.2
3.1 Interiace-matching loss
To maintain physical consistency across adjacent subdomains, continuity is enforced by applying matching Dirichlet boundary conditions at the interfaces. The interface-matching loss is defined as:
NM
LM =
Σ (Fin - F2,n
(2)
NM
where Lu represents the mean squared error (MSE) of field variables between adjacent subdomain interfaces, Fi and F2, across NM matching points. This formulation ensures the continuity of fluid flow variables (e.g., velocity and pressure) across subdomain interfaces. It is important to note condition constrains only the geroth-order continuity. Improved
that the current continuity and physical consistency can be achieved by extending the matching conditions to include higher-order derivatives, such incorporating first-order derivatives of the flow variables (e.g.. volumetric flux). However, further improvements by including higher-order derivatives incur the cost of increased computational complexity.
3.2 Global mass conservation loss (MDPINN-GD)
Whilst subdomain and interface predictions are greatly improved with MDPINN, there is still tendency to violate global mass conservation when applied to complex 3D fluid systems. To mitigate this, we introduced an additional bhasec constraint into the loss function enforce global volumetric flow rate conservation. Specifically, a global continuity loss term, denoted as Lv. is incorporated to penalize discrepancies between the total volumetric inflow and outflow across the entire fluid network. To maintain scale invariance and improve numerical stability during training, all flow rates are normalized against a reference volumetric flow rate, Ug where A, wd the cross-sectional area at the inlet. The volumetric flow rate loss was formulated as:
Nen
Novt
Lv
Σ (u (u-n):Ar
Σ (u 7)m Am
Ao
where u is the local velocity vector, n is the unit normal vector to the boundary surface, and A, and Am represent the discretized cross-sectional areas perpendicular to n the inlet and outlet boundaries, respectively. Nin and Nout correspond to the number of discrete inlet and outlet segments used in the numerical integration.
3.3 Composite loss function
The total loss function is constructed a composite of multiple components: the sum of PDE residual loss, boundary condition loss, the interface-matching loss across all subdomains, and global mass
5

---

conservation. Thus, the composite loss function, L, to minimize is written as:
NR
No NBC
NF 2 1is + Lv
L NRNR
NS
50,n
NF
(4)
where summation is performed across N5 =9 subdomains and NF 8 interfaces. The number of collocation points, boundary points, PDE residual loss, MDPINN-GD predictions on the boundary, and boundary condition ground truths are represented by NR NBC, e, S and Ei.. respectively. Factors leading the double summations arise due to the MSE function. Additional details on traditional PINN loss functions be found in the seminal work by Raissi et al. [
4 Results and discussion
4.1 Reference flow patterns
Simulated streamlines in three different planes, shown in Fig. (a-c), illustrate the expected flow patterns in this 3D cross-flow channel. In the (a) zy-plane, the flow remains nearly straight until the intersection, where it is deflected into the perpendicular outlet branches, and recirculating flow near the outlet walls is clearly visible. In the (b) yz-plane at the entrance of the outlet section (subdomain 4), the flow moves upward, deflects the top, and forms recirculating zones near the sidewalls. In the (c) yæ-plane at the exit of the outlet section (subdomain 5), strong flow mixing occurs, with a prominent recirculation zone forming at the channel center. Overall, the simulation captures the key transport features of 3D cross-flow flow junction, including laminar inlet development, intersection-in -nnduced deflection, and outlet mixing and recirculation. The CFD simulation results in Fig. 2 (d) further highlight the overall flow patterns, particularly in the ry-plane view at 0.5 (mid-plane). the inlet boundaries located on the left and bottom, the flow exhibits classical Poiseuille flow features, with maximum stream- stream-wise velocity at the center and decreasing velocity near walls. contrast, the outlet boundaries on the top and right display more complex flow behavior due to the merging and redirection of streams.
(a)
(b)
1.0
Streamlines
-.5
1.0
(d)
Pressure
10 [Pa]
x-velocity
[mvig]
y-velocity
[mrs]
z-velocity
10 10m(mm/)
2
1.0
Simulstion
0.5
-2.5
-7.5
0.0
10P [mvs]
[m/s]
10" [mva]
minglennttworkeeo PINN
2
2
1.0
2.5
0.5
-2.5
7.5
0.0
Figure 2: Streamline plots for simulation at: (a) zy-plane 0.5, (b) inlet yz-plane of subdomain 4 at 0.6, and (c) outlet yz-plane of subdomain at x = 1. These views demonstrate overall flow patterns and the change flow profile moves downstream. Comparison of pressure, x-velocity, y-velocity, and z-velocity fields between the (d) CFD simulation and (e) single-network PINN model the 0.5, viewed in the zy-plane.

---

4.2 Single-network PINN
To understand the limitations of single-network PINNs for modeling 3D fluid networks, we first implemented and tested single-network architecture without domain decomposition. The network was optimized through hyperparameter tuning based on previous studies B3}. The architecture consisted of fully connected network with 10 hidden layers and 1 1.50 neurons per layer, employing adaptive sine activation functions. Training was conducted with 250,000 collocation points (average separation of 2.444 um between points) and boundary points sampled uniformly along a grid with spacing of J.m. The network was trained using the same two-stage optimization (Adam + LL-BFGS) as the MDPINN. However, the model produced highly inaccurate predictions, shown in the zy- plane view in Fig. 2 (e), with additional plane views provided in Appendix [A.3] The network failed to capture the expected flow patterns shown in the simulation results: fluid entered only from the left inlet and dissipated rapidly along the channel walls, without meaningful circulation or cross-flow behavior. Notably, even with low final loss (on the order of I 1), the ängle-networt PINN was unable to resolve the flow features of the intersecting channel geometry. Additional experiments of various geometries shown in Appendix A.
To quantitatively assess the model accuracy, we evaluated two error metrics: the mean absolute error, c, and the maximum absolute error, Caxx as given in Eq. where $,, L and N, represent the PINN prediction, simulation result, and number of nodes in simulation, respectively. For the single-network PINN, for pressure, z-velocity, y- y-velocity, and z-velocity were 3.41 kPa, 0.52 m/s, 0.52 m/s, and 0.18 m/s, respectively. The corresponding émax were 23.6 kPa, 2.16 m/s, 2.16 m/s, and 1.79 m/s, respectively. This indicates mean errors of nearly 20% of Up for the x- and y-velocity components. Furthermore, the maximum velocity errors reach nearly 81% of Up. suggesting that the discrepancies in the single-network PINN are on the same order of magnitude as Uo of the system, which is highly undesirable as errors will impose similar influence on the flow dynamics as Ug.
p
P
Cmax max (Ø. Po.sl)
(5)
4.3 MDPINN-GD
To identify the most effective optimization strategy for the MDPINN-GD framework, we compared
three approaches in terms of their accuracy and computational efficiency. The approaches included: (1) MDPINN,, a baseline model trained with 20,000 collocation points (average separation of
2. 729 um between points) per subdomain; (2) MDPINN,, higher-resolution model using 40,000 collocation points (average separation of 2.165 μm between points) per subdomain; and (3) MDPINN- GD, an optimized model using 20,000 collocation points per subdomain incorporating global mass conservation constraint. The training loss plot can be found in Appendix A..4 The training times for MDPINN, 2DPINN,. and MDPINN-GD were 24.3 hours, 38.8 hours, and 68 hours, respectively. Corresponding GPU memory usage was approximately 40 GB, 60 GB, and 45 GB, All models were trained on a single Nvidia A100 80GB GPU.
The field predictions from the three MDPINN networks are shown in Fig. 3 MDPINN, failed to satisfy the mass continuity constraint, as evidenced by zero outlet flows in the y-velocity field. Additionally, the pressure at the bottom-face inlet was lower than that at the right-face outlet, contradicting the expected pressure gradient from inlet to outlet. However, compared single- network PINN, the results indicate that MDPINN outperforms even with fewer collocation points. MDPINN2, which utilized doubled collocation point density, showed improved mass continuity and captured general flow patterns. This suggests that increasing the number of collocation points can enhance prediction quality, although incurs additional computational cost. However, MDPINN, still underestimated both velocity magnitudes and pressure values in the outflow region, implying that greater collocation point density alone was insufficient for accurate field prediction regions of complex flow behavior. These results suggest that using nulti-network architecture without explicit enforcement of global continuity is insufficient for accurately predicting complex flow fields. In contrast, MDPINN-GI successfully demonstrated mass continuity and accurate magnitudes for all flow variables throughout the domain. A further comparison the channel cross-section (yz=-lane view) at the entrance of the outlet section (subdomain 4) revealed that MDPINN-GD accurately captured both z- and y- yeleocitty fields, whereas MDPINN, significantly underpredicted the y-velocity relative to the CFD results.
7

---

Pressure
xivelacity
y-velocity
z-velacity
1.0
Secction y2 plane Intat
Simulation
0.5
0.0
MDPINN,
0.5
1.0
MDPINN
0.0
MDPINN-COM GiD
0.5
0.0
1.0 0.0
0.5
0.0
0.5
1.0
0.0
0.5
1.0
06
10
0.0
0.5
Figure 3: Pressure, zz-elooityy y-velocity, and z-velocity field comparisons of CFD simulation and MDPINN models of zy-plane at 0.5 (mid-plane). Additional views for and y-velocity of ye-plane at the inlet of subdomain is also shown to highlight 3D features.
To quantitatively compare the predictive accuracy, we evaluated and émax across all four fields. The MDPINN-GD achieved mean improvement of 76.2% and 52.9% over MDPINN, and MDPINN,, respectively. Similarly, the mean 6max improvement of MDPINN-GD over MDPINN, and MDPINN, was 75.2%, and 49%, respectively. These results indicate that improvements observed evenly across and Cmax- From MDPINN, to MDPINN,, comparison shows that doubling the collocation points results in approximately 50% decrease in error, which yields to comparable flow field predictions. However, clearly shows that incorporating global constraints provides further substantial accuracy gains without additional collocation points. Overall, MDPINN-GI achieves roughly twice the accuracy of MDPINN,. These results underscore the importance of combining domain decomposition with global continuity enforcement and sufficient sampling to achieve robust and physically consistent PINN predictions in complex 3D flow configurations. Error comparisons for all PINN models can be found in Appendix A.55
To understand how the complexity fluid dynamics affects model accuracy, e and émax were evaluated for each subdomain, shown in Table [ A key observation is that the largest streamwise errors (i.e., errors aligned with the main flow direction) occurred in the outlet subdomains, where flow deceleration and recirculation introduce additional physical complexity. The largest x and y-velocity error, (U) and 6 (W), is 29.9 10 m/s, corresponding to 14.9% of Un- Notably, the largest z- velocity and pressure errors, e (W) and (P), are observed the cross junction (subdomain 3), where converging inlet flows subsequently diverge, creating strong velocity gradients in all directions. Complete flow fields the inlets and outlets of subdomain 3 are shown in Appendix A.6 Analysis of and Ema shows similar trends, with slight deviations of symmetry in the values. For example, comparing Cmax along the streamwise directions in subdomains 1, 5, 6, and 9 (inlets and outlets) reveals that this asymmetry is likely due to numerical randomness rather than physical effects. Additional error metrics normalized by inlet velocity and inlet dynamic pressure are provided in Appendix A.7
The MDPINN-GD results further illustrated in Fig. which presents five views of the x-velocity field alongside corresponding predictions and absolute error distributions. In the ry-plane, error concentrations appear specifically at the diagonal intersection of the channels and the right outlet, likely due to intense mixing and sharp changes in velocity direction. Increased errors are also observed at the inlets of subdomain 4 (yz-plane) and subdomain (sz-planee, particularly near the top and right sides, respectively. These regions correspond to the top-right corner of the channel intersection, where the fluid undergoes sharp directional changes. potentially leading localized numerical singularities. In addition, the outlet regions of subdomain 5 (yz-planc) and subdomain

---

Table Subdomain and Gnax for MDPINN- GD with bold indicating maximum across each row.
Left inlet
Right outlet Intersect
Bottom inlet
Top outlet
Subdomain number
5
6
8
(P)
[10 Pa]
1.95
1.04
16.5
13.9
2.15
3.31
1.13
14.5
2.25
(C)
[10-2 m/s]
9.65
18.9
21.5
24.4
29.9
1.96
2.64
11.7
8.16
(V)
[10-2 2 m/s]
1.85
2.56
21.4
11.7
8.21
9.84
18.9
24.4
29.7
(W)
[10-2 m/s]
2.79
3,46
7.99
6.88
6.27
2.92
3,46
6.92
6.16
6max (P)
[10* Pa]
74.2
12.2
25.4
22.2
4.96
85.1
10.2
23.2
4.96
Emax
[m/s]
1,51
0.702
1.04
1,41
1.54
0.39
0.31
0.69
0.44
6max
(V)
mm/s]
0.43
0,41
1.04
0.67
0.44
1.72
0.69
1.47
1.60
6max (W)
[10
m/s]
4.28
2.96
6.39
5.49
4.46
4.14
3.04
5,49
4.52
Xy-plane
plane
5
(mve])
[mAg]]
Im/el
Im/s
2.5
0.5
1.5
0.5 y
0.5 2
0
10
1.0 [mvi)]
[mvis]
1.0 [m/s]
(mvie)
1.0
[mve]
2.5
MDP
y
-
0.0
0.0
0.0
1.0 [mvi]]
[m/h]]
1.0 (md)]]
[mb)]]
[mss]
error
0.5
0.5
0.5
0.5
0.2
0.0
10
10'
1.0
0.5
1.0
Figure 4: Comparison of z-velocities between CFD simulation and MDPINN-GD with absolute errors. Viewed at: zy-plane at = = 0.5 (mid-plane), yz-plane at inlet of subdomain 4, yz-plane at outlet of subdomain 5, xz-plane at inlet of subdomain 8, and zz-plane at outlet of subdomain 9.
(zz-plane) exhibit asymmetric errors near the walls, likely resulting from the propagation of numerical artifacts originating at the corners. These artifacts may accumulate along the flow path, contributing to the final prediction errors. Additional visualization and error distributions can be found in Appendix A.6
5 Conclusion
In summary, this work addresses key limitations in applying PINNs to the modeling of fluid dynamics in complex geometries, i.c., intersecting 3D fluid channels with high aspect ratio. Traditional PINNs often struggle with convergence and predictive accuracy in high-aspect-ratio geometries, which induce intricate flow features such as deflections, mixing, and recirculations. To overcome these challenges, we first demonstrated that vanilla MDPINN that leverages domain decomposition to facilitate localized learning, MDPINN,, was able to outperform the single-network PINN with 39 % less collocation points while decreasing error by 90.2%. Next, we presented an enhanced MDPINN- GD framework that enforces global dynamics by integrating global mass conservation constraints into the loss formulation. Together, these enhancements led predictive accuracy over the single-network PINN and MDPINN architectures without the global mass conservation. Furthermore, mean absolute velocity errors of MDPINN-GD were below 14.9% when normalized by the inlet velocity. This indicates that the proposed method is able to accurately predict the complex flow phenomena, with minimal errors likely caused by the propagation of corner singularities. Overall, the proposed approach effectively captures complex cross-flow interactions and resolves steep inlet pressure gradients, demonstrating its potential as scalable, tigh-fidelity and physically consistent surrogate modeling tool for fluid simulations.
22.9%-77.88% improvement in average

---

Limitations Despite these advancements, several limitations remain. As shown in Fig. 3 the predicted z-velocity exhibits noticeable deviations from the CFD baseline, suggesting a need for finer spatial resolution or improved learning of secondary flow features. This issue underscores the necessity for future research into adaptive loss weighting strategies, multi-scale network architectures, enforcing boundary conditions as hard constraints, and enhanced interface treatments for domain coupling. Furthermore, the demonstrated method has so far been applied to a 3D intersecting channel flow; experiments other geometries can be helpful to assess its robustness and generalizability. Future research will focus on advancing the MDPINN-GD framework through more adaptive and generalizable learning strategies on various geometries. Additionally, we plan extend the framework to handle transient and multi pulti-physics systems, thereby demonstrating the scalability of PINNs in dynamic and coupled environments. We also acknowledge that systematic benchmarking of inference speed across various discretization levels are necessary to justify computational efficiency claims. We consider this an important direction for future work.
Acknowledgments and Disclosure of Funding
We would like to recogmize Nagahiro Ohashi and Prof. Beomjin Kwon (School for Engineering of Matter, Transport and Energy Arizona State University) for their substantial intellectual contribu- tions to this work. They have led and were deeply involved in the research design, execution, and preparation of the manuscript. Due to an unintentional clerical error during the submission process, they are not listed as co-authors (Nagahiro Ohashi as the first author and Prof. Beomjin Kwon as the second author) of this NeurIPS version, but they should be regarded as key contributors to the ideas and results presented in this paper. This work was supported by the National Science Foundation grant under Grant No. 2337973. Additionally, this research was partially supported by the Nano & Material Technology Development Program through the National Research Foundation of Korea (NRF), funded by Ministry of Science and ICT (RS-2024-0044863)) and Arizona State University startup funds.
References
[1] M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informedd neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686 707, February 2019.
[2] Salvatore Cuomo, Vincenzo Schiano Di Cola, Fabio Giampaolo, Gianluigi Rozza, Maziar Raissi, and Francesco Piccialli. Scientific Machine Learning Through Physics-Informer Neural Networks: Where we are and What's 's Next. Journal of Scientific Computing, 92(3):88, July 2022.
[3] Zaharaddeen Karami Lawal, Hayati Yassin, Daphne Teck Ching Lai, and Azam Che Idris. Physics-Informer Neural Network (PINN) Evolution and Beyond: A Systematic Literature Review and Bibliometric Analysis. Big Data and Cognitive Computing, 6(4):140, December 2022. Number: 4 Publisher: Multidisciplinary Digital Publishing Institute.
[4] Solji Choi, Ikhwan Jung, Hacun Kim, Jonggeol Na, and Jong Min Lee. Physics-informet deep learning for data-driven solutions computational fluid dynamics. Korean Journal of Chemical Engineering, 39(3)551--528 March 2022.
[5] Shengze Cai, Zhicheng Wang, Sifan Wang, Paris Perdikaris, and George Em Karniadakis, Informe- Neural Networks for Hcat Transfer Problems. Journal of Heat Transfer, 143(6):06000,, June 2021.
[6] Xiao-dong Bai, Yong Wang, and Wei Zhang. Applying physics informed neural network for flow data assimilation. Journal of Hydrdynamics, (006):05501000 December 2020.
[7] Yikun Yang. Xifeng Wang. Jinfeng Li, and Riletu Ge. data- dhytic driven method for gear fault diagnosis using PINN and sacudo-dynamic features. Measurement, 236: 15124, August 2024.
[8] Félix Fernández de la Mata. Alfonso Gijón, Miguel Molina-Solana Solana, and Juan Gómez- Romero. Physics- informed neural networks for data-driven simulation: Advantages, limitations, and opportunitics. Physica A: Statistical Mechanics and its Applications, 18:11,, January 2023.
[9] Lu Zhu, Xianyang Jiang, Adrien Lefauve, Rich R. Kerswell, and P. F. Linden. Physics-informed neural net- work to augment experimental data: an application to stratified flows, September 2023. arXiv:2309. 14722
[physics].
10

---

[10] Hesheng Tang, Yangyang Liao. Hu Yang. and Liyu Xie. A transfer learning piaiiinggg-yisssssse informed neural network (TL-PINN) for vortex-induced vibration. Ocean Engineering, 13101, December 2022.
[11] Konstantinos Prantikos. Stylianos Chatzidakis, Lefteri H. Tsoukalas, and Alexander Heifetz. Physics- informed neural network with transfer learning (TI.-PINN) based on domain similarity measure for prediction of nuclear reactor transients. Scientific Reports, 13(1):16840) October 2023. Publisher: Nature Publishing Group.
[12] Nagahiro Ohashi, Leslie K. Hwang, and Beomjin Kwon. Physics-informed neural networks for multi-field visualization with single-color laser induced fluorescence. Al Thermal Fluids, 1:100005, March 2025.
[13] Vikas Dwivedi and Balaji Srinivasan. Physies Informed Extreme Learning Machine (PIELM) A rapid method for the numerical solution of partial differential equations. Neurocomputing, 391:96- 118, May 2020.
[14] George Em Karniadakis, Ioannis G. Kevrekidis, La Lu, Paris Perdikaris, Sifan Wang, and Liu Yang. Physics-informed machine learning. Nature Reviews Physics, 3(6):222-44) 440, June 2021.
[15] Xiaowei Jin, Shengze Cai, Hui Li, and George Em Karniadakis. NSFnets Wavier-Stokee flow nets): Physics-infonee informed neural networks for the incompressible Ntvier-Stokes equations. Journal of Computa- tional Physics, 426: 109951, February 2021.
[16] Wei Xuan Chan, Wenhao Ding, Binghuan Li, Hong Shen Wong, and Choon Hwai Yap. Role of physics- informed constraints in real-time estimation 3D vascular fluid dynamics using multi-case neural network. Computers in Biology and Medicine, 190: 1 10074, May 2025.
[17] Philip Heger, Daniel Hilger, Markus Full, and Norbert Hosters. Investigation physics-informed deep learning for the prediction of parametric, tiree-dimensional flow based on boundary data. Computers & Fluids, 278: 106302, June 2024.
[18] Gaurav Kumar Yadav, Sundararajan Natarajan, and Balaji Srinivasan. Distributed PINN for Linear Flasticity
A Unified Approach for Smooth, Singular, Compressible and Incompressible Media. International Journal of Computational Methods, 19(08):214200 October 2022. Publisher: World Scientific Publishing Co.
[19] Philipp Moser. Wolfgang Fenz, Stefan Thumfart, Isabell Ganitzer, and Michael Giretzlehner. Modeling of 3D Blood Flows with Physics-Informed Neural Networks: Comparison of Network Architectures. Fluids, 8(2):46, Fehruary 2023. Number: Publisher: Multidisciplinary Digital Publishing Institute.
[20] Aditi Krishnapriyan, Amir Gholami, Shandian Zhe, Robert Kirby, and Michael W Mahoney. Characterizing possible failure modes in physics-infomed neural networks. In Advonces Neural Information Processing Systems, volume 34, pages 26548 26560. Curran Associates, Inc., 2021.
[21] Linyan Gu, Shanlin Qin, Lei Xu, and Rongliang Chen. Physics-infommnt neural networks with domain decomposition for the incompressible Navier Stokes equations. Physics of Fluids, 36(2):22114, February 2024.
[22] Khemraj Shukla, Ameya D. Jagtap, and George Em Karniadakis. Parallel innsics-informed neural networks via domain decomposition. Journal of Computational Physics, 447:1 10683, December 2021.
[23] R. Laubscher. Simulation of multi- flow and heat transfer using physics Physics of Fluids, (3870001 August 2021.
informed physics-informed neural networks.
[24] Arshia Merdasi, Saman Ebrahimi, Xiang Yang, and Robert Kunz. Physics Informed Neural Network application mixing and heat transfer in combined electroosmotic-presttee driven flow. Chemical Engineering and Processing- - Process Intensification, 193:109540, November 2023.
[25] Vikas Dwivedi, Nishant Parashar, and Balaji Srinivasan. Distributed physics informed neural network for data-efficient solution to partial differential equations. Neurocomputing. 420:299-316, January 2021. arXiv:1907 N0w:1997708967 [es].
[26] Ameya D. Jagtap, Ehsan Kharazmi, and George Em Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems. Computer Methods in Applied Mechanics and Engineering, 365:1 1 13028, June 2020.
[27] Ameya D. Jagtap and George Em Kariaadaiis Extended Physics-Informer Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations. Communications in Computational Physics, 28(5), November 2020. Institution: Brown Univ., Providence, RI (United States) Publisher: Global Science Press.
11

---

[28] Shibo Li. Michael Penwarden, Yiming Xu, Conor Tillinghast, Akil Narayan, Robert M. Kirby, and Shandian Zhe. Meta Learning of Interface Conditions for Multi-Doman Physics-Informed Neural Networks. In Proceedings of 40th International Conference on Machine Learning, pages 19855-19881, July 2023.
[29] Michael Peawarden, Ameya D. Jagtap, Shandian Zhe, George Em Karniadakis, and Robert M. Kirby. unified scalable framework for causal sweeping strategies for Physics-Informee Neural Networks (PINNs) and their temporal decompositions. Journal of Computational Physics, 493:112466 12464, November 2023.
[30] Sifan Wang. Yujun Teng. and Paris Perdikaris. Understanding and Mitigating Gradient Flow Pathologies Physics-Informnt Neural Networks. SIAM Journal on Scientific Computing, 43(5):A3055 A3081, January 2021. Publisher: Society for Industrial and Applied Mathematics.
[31] Wei-Liem Loh. On Latin hypercube sampling. The Annals of Statisticx, 24(5):2058- 2080, October 1996. Publisher: Institute of Mathematical Statistics.
[32] Ameya D. Jagtap, Kenji Kawaguchi, and George Em Karniadakis. Adaptive activation functions accelerate convergence in deep and thysics-infomat neural networks. Journal of Computational Physics, 404:109136. March 2020.
[33] Shengze Cai, Zhiping Mao, Zhicheng Wang, Minglang Yin, and George Em Kariaaaaiis Physics- informed neural networks (PINNs) for fluid mechanies: review. Acta Mechanica Sinica, 37(12):1727-177) -7388 December 2021.
12

---

A Technical Appendices and Supplementary Material
A.1 Illustration of PINN failures
Here we show some typical failure points of single-network PINNs, which demonstrates the diffi- culties of training on non-canonical geometries. A 2D straight channel evaluated two different aspect ratios are shown. The pressure fields are good indicators of poor network performance, since typically the hardest to resolve. Fig. A.I (a) and (b) show channel aspect ratio of 1:5 and 1:10 (width:length), respectively. It is clearly visible that these high aspect ratios pose an intrinsic challenge due multi-scale issues. Next, we demonstrate a 3D cubic channel domain. Fig. A.I (c) shows the yæ-plane view PINN predictions at the outlet, in which asymmetry along the 2-axis is observed. The likely sources of error are corner singularities that PINNS struggle approximate.
PINN Pressure
Simulation Pressure
AR-1-5
1.0
AR: Aspact ratio (width:length)
0.5 y:
0.0
By
2y
3y
Ay
by
(b) AR-1:10
y
0.0
10y
2y
8y
By
10y
Pressure
X-Velocity
y-velocity
2-velodity
(c) 3D
0.5
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Figure A.I: Demonstration of 2D straight channel PINN and simulation pressure field with aspect ratios (width:length) of (a) 1:5 and (b) 1:10. 3D cubic channel flow fields at the outlet are shown in (c) to demonstrate effects of corner singularities.
13

---

A.2 MDPINN-GD pseudocode
Algorithm MDPINN-GD with Ng domain decomposition
Input: Local spatial coordinates: 2:
Output: Local fluid parameters: P. U,
1: Create Ng subdomain geometries
2: Define subdomain neural network architecture: layers 4, neurons
3: Define activation function (i.e., adaptive sine)
4: Define NR collocation points and boundary points
5: Define fluid properties, boundary conditions and initial conditions (if any)
6: Define matching conditions and global continuity constraints
7: for 0 to Ns do
8: Assign to x n fully-connected neural network
9: Initialize network weights and bias: W~Xavver normal, b = 0
10: end for
Initialize loss values: LR. LBC. LM =0
12: Load W and b into Adam and L-BFGS optimizer
max max_epoch do 14: for = 0 to Ng do
for
Extract thh subdomain geometric and fluid properties
16:
Compute fluid parameters on all coordinates with forward propagation Calculate PDE loss for 5th subdomain, T Add L to total LR
18:
19:
Extract tth subdomain boundary conditions Calculate boundary loss for th subdomain, LBC Add BC to total LBc
21:
22:
end for
23:
for f to Np do
Extract pairs of subdomain matching conditions
25:
Calculate matching loss for 4h subdomain pair, A
26:
Add L to total Las
27:
end for
28:
for = 0 to num num_inlets do
29:
Calculate volumetric flow rate at inlet face i, Qi
30:
Add Q to total Qin
31:
end for
32:
for 0
outlets do Calculate volumetric flow rate at outlet face o, Qout
33:
34:
Add Qo to total Qout
35:
end for
36:
Calculate global continuity loss, Lv= (Qout Qin)
37:
Calculate total loss, L = La
+ Loc
Lu
+ Lr
No
No
38:
if Adam max_epoch then
39:
Reset Adam optimizer gradient
40:
Calculate 30 with backpropagation
41:
Update Adam optimizer
42:
else
43:
Reset L-BFGS optimizer gradient
0L Calculate ae and 0² L with backpropagation aga
45:
Update L-BFGS optimizer
46:
end if
47: if L < convergence_criteric then
End training
49:
end if
50:
end for
14

---

A.3 Model comparisons
Comparisons single-network PINN, 1DPINN1, 2DPINN2, and MDPINN-GH for various views are given in the following figures. Flow fields are shown at the outlets of subdomains 5 and 9, and inlets of subdomains and 8.
Simulation
Singfenentvrkk PINN
MDPINN
MDPINN:
MOPINN-GD
1.0 [Ps)
anssed
0.5
0.0
1.0
Aypoyeaxx
-0.5 V
0.0
1.0
10 [mpe]
y
10" [m/]]
Aypopozz
y
0.0
0.5
1.0
6.5
1.0.0.0
0.5
1.000
0.5
1.000
0.5
1.0
Figure A.2: Comparison of pressure, x-velocity, y-velocity, and z-velocity fields between the simulation and PINN models at outlet of subdomain viewed in the yz-plane.
Simulation
Singte-network
MDPINN:
MDPINN,
MOPINN-GO
[Ps]
Prossura
0.6
0.0
[m]]
000
..0
1.0
[m/s]]
0.5
0.0
1.0
[mm]]
0.6
0.0
1.000
1.00.0
0.
1,000
1.0000
Figure A.3: Comparison of pressure, v-velocity, y-velocity, and z-velocity fields between the simulation and PINN models at outlet of subdomain 9, viewed in the æz-plane.
15

---

Bingle-netiorkk
MOPINN,
MDPINNN
Presuuee
[m/)]
A1000
2.5
[msr]]
-0.5
0.0
0.5
0.0
0.0
0.5
0.5
1,0000
1.00.0
Figure A.4: Comparison of pressure, z-velocity, y-velocity, and zevelocity fields between the simulation and PINN models at inlet of subdomain viewing the ye-plane.
MDPINN,
1.0
Pressure
[m]]]
[ndk]]
2.5
0.5
0.0
[mm]]
0.5
0.0
Figure A.5: Comparison of pressure, z-velocity, y-velocity, and z-velocity fields between the simulation and PINN models at inlet of subdomain 8, viewed in the zl-plane.
16

---

Simulation
MDPINN,
1.0
Preasume
[mv]]]
0.5
0.0
[mw]]
-0.5
.000
110
-0.5
0.0
1.0.000
1.00.0
0.5
1.0
1.0000
Figure A.6: Comparison of pressure, z-velocity, y-velocity, and z-velocity fields between the simulation and PINN models at the center of the cross-shaped channel, viewed in the zy-plane.
A.4
Loss plot
To understand the training dynamics, the evolution of loss components for MDPINN-GD is shown Fig. A.7 Fig. A.7 (a), the total loss gradually decreases, with the most contribution from orders of magnitude lower contributions from the PDE loss and global continuity loss. Notably, the global continuity loss decreases rapidly by several orders of magnitude from around 1,000 epochs among the 100,000 epoch and continues to drop below 10 10-8 indicating that the PINN effectively enforces inter-network continuity across subdomains. Meanwhile, the boundary loss decreases more slowly and remains the dominant contributor the total loss, suggesting that enforcing boundary conditions in as hard constraints may be necessary. Fig. A.7 (b) further decomposes the PDE loss into its physical components. Initially, all components exhibit rapid сттог reduction, followed by more gradual convergence. Interestingly, the z-momentum loss reaches lower final values compared the other losses, reflecting the geometric anisotropy of the domain, where flow variations in the z-direction are less pronounced, leading to faster convergence in that component.
the boundary loss and
(a)
10:
(b)
Continulty loss x-momentum
10a
10
y-momentum loss z-momentum loss
10-
10"
Los8
10
10
Total loss
Boundary
PDE loss
10:90
Global continulty loss
104
10P
10
10c
10a
10
10a
10"
101
102
107
10*
10
Epochs
Epochs
Figure A.7: Loss plots of MDPINN-GD: (a) high-level loss terms: total, boundary condition, PDE, and global continuity loss, (b) PDE loss terms: continuity, z-momentum, y-momentum, and z- momentum.
17

---

A.5 Model error comparison
Table A.I: Mean and max absolute errors for all PINN models. MDPINN models are averaged across all subdomains. Overall mean errors averaged across all fields and percentage improvement of MDPINN-GD are compared against other models.
Single-network PINN
Mean errors
MDPINN-GD
MDPINN,
MDPINN2
(P)
[kPa]
0.62
3,41
3.55
1.76
c (U)
[m/s]
0.14
0.52
0.7
0.33
(V)
[m/s]
0.14
0.52
0.96
0.33
e (W)
[m/s]
0.052
0.18
0.12
0.076
<max
[kPa]
1.63
23.6
8.11
3.4
<max
[m/s]
0.3
2.16
1.52
0.66
[m/s]
0.3
2.16
2
0.68
max
Emax
(W)
[m/s]
0.08
1.79
0.18
0.12
MDPINN-GI mean improvement
74.8%
76.2%
52.9%
MDPINN-GD mean Emax improvement
90.2%
75.2%
49%
18

---

A.6 MDPINN-GD error
Comparison of MDPINN-ON GD predictions the CFD groundtruth with absolute error fields for various views which include: left and bottom inlets of subdomain 3, right and top outlets of subdomain 3, outlet of subdomains 5 and 9, and zy-plane view 0.5 (mid-plane).
Pressure
x-velocity
y-velocity
z-velocity
[Pa]
[m/i]
[nve]
1.0 [m/s]
1.25
Simulation
0.75
10
0
0.0
10"
[m/s]]
[m/s]
1.0 [mvs]
MOPINN-GD
0.75
-0.5
-2
10a
0.0
[Pa]
[m/s]
[m/s]
1.0 [m/s]
Absolute
0.5
10
0.0
1.0
0.5
0.0
1.0
0.0
1.0
Figure A.8: Pressure, x-velocity, y-velocity, and z-velocity field comparisons of CFD simulation and MDPINN-GD at left inlet of subdomain with absolute error field viewing the y2 plaplane.
Pressure
x-velocity
y-velocity
z-velocity
[Pa]
[m/s]
[m/s]
1.0 [m/s]
1.25
Simulation
0.75
0.5
-2
10
0
10"
[Pa]
[m/s]
[m/a]
1.0
[m/s]
MDPINN-GD
1.25
0.5
10
0.0
10"
[Pa]
[mva)
[mva]
1.0
[m/s]
Absolute error
8
6
0.5
2
0.5
0.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Figure A.9; Pressure, z-velocity, y-velocity, and z-velocity field comparisons of CFD simulation and MDPINN GD at bottom inlet of subdomain with absolute error field viewing the æ2-
19

---

Pressure
x-velocity
y-velocity
z-velocity
[Pa]
[m/s]
[m/s]
[m/s]
2.5
Simulation
-
10
0.0
10-
[Pa]
[m/s]
[m/e]
[m/s]
MDPINN-GD
2.5
1.5
0.5 y
10
[Po]
[m/s]
[m/s]
1.0 [m/s]
Absolute error
2
1.5
y
0.5
10%
10a1
10a
10a
Figure A.10: Pressure, z-velocity, y-velocity, and z-velocity field comparisons of CFD simulation and MDPINN-GD at right outlet of subdomain with absolute error field viewing the yz-plane.
Pressure
x-velocity
y-velocity
z-velocity
[Pa]
[m/e]
[m/s]
[m/a]
Simulation
2.5
0.5
0
-5
10
0.0
10a1
(Pa]
[m/a]
[m/e]
1.0
[m/s]
MDPINN-OD
2.5
1.5
0.5
0.0
101
[Pa]
[m/e]
[m/e]
1.0
[m/a]
2
8
1.5
Absolute
0.5
10P
10a1
10a
10a
x
x
Figure A.11: Pressure, zeveoocity y-velocity, and z-velocity field comparisons of CFD simulation and MDPINN-GI at top outlet of subdomain with absolute error field viewing the z2 zz-plane.
20

---

Pressure
x-velocity
y-velocity
z-velocity
[Pa]
[m/s]
[m/s]
[m/s]
Simulation
-2
10
0.0
10-
[Pe]
Im/e]
[m/s]
[m/s]
2
MDPINN-GD
0.5 y
[Po]
[m/s]
[m/s]
1.0 [m/s]
Absolute error
1.5
y
0.5
10a
10a
Figure A.12: Pressure, z-velocity, y-velocity, and z-velocity field comparisons of CFD simulation and MDPINN-GD at outlet of subdomain 5 with absolute error field viewing the yz-plane.
Pressure
x-velocity
y-velocity
z-velocity
[P]
[m/s]
[m/s]
[m/e]
Simulation
2
-4
-6
-2
10
0.0
10
[Pa]
[m/a]
[m/s]
1.0 [m/s]
MDPINN-CO
2
0.5
0.0
[Pa]
[m/e]
[m/s]
1.0 [m/e]
Absolute error
0.5
10-1
10a
X
X
Figure A.13: Pressure, z-velocity, y-velocity, and z-velocity field comparisons of CFD simulation
and MDPINN- GD at outlet of subdomain with absolute enrr field viewing the x2-plane.
21

---

Pressure
x-velocity
y-velocity
z-velocity
[Pa]
[m/s]
[mve]
[m/s]
5
Simulation
2.5
y
10a
0.0
10a
[Pa]
[m/s]
[m/s]
1.0 [m/s]
2
MDPINN-CO
0.5 y
-2.5
-7.5
10
[Pa]
[m/s]
[m/s]
1.0
[m/s]
Absolure error
2
0.8
0.6
y
0.2
0.2
10
10*
x
Figure A.14: Pressure, z-velocity, g-velocity, and z-velocity field comparisons of CFD simulation and MDPINN- at center of cross-shaped channel with absolute error field viewing the ry ryy-lanne
A.7 MDPINN-GD normalized absolute error
Table A.2:
Subdomain mean and max absolute errors for MDPINN-GD with bold indicating maxi- mum across each row. Values are given percentage of reference inlet velocity (U)) and reference inlet dynamic pressure (pnUg).
Left inlet
Intersect
Right outlet
Bottom inlet
Top outlet
Subdomain number
2
3
4
5
6
8
9
% (P)
4.86
2.59
40.83
34.7
5.36
8.26
2.81
36.2
5.62
% (U)
4.82
9.43
10.7
12.1
14.9
0.98
1.31
5.86
4.07
% (V)
0.92
1.27
10.7
5.83
4.10
4.92
9.48
12.18
14.8
% (W)
1.39
1.73
3.99
3.43
3.13
1.46
1.72
3.45
3.07
% 6mxx
(P)
185.4
30.7
63.6
55.6
12.3
212.6
25.5
58.0
12.4
% Emax
(U)
75.6
35.1
52.1
70.5
76.8
19.5
15.9
34.7
22.1
% max
21.6
20.7
52.0
33.8
22.2
85.9
34.5
73.5
80.1
%c0max
(W)
21.3
14.8
31.9
27.4
22.3
20.7
15.2
27.4
22.6
22

---

A.8 Loss weight ratio tuning
The loss weight ratio between boundary loss and PDE loss were explored for the optimal MDPINN- GD result. The z-velocity field predictions at different views are shown in Fig. [A.15] in which 10x PDE loss, 20x PDE loss, and 10x boundary loss were compared against MDPINN- (uniform loss). It is evident that uniform loss is optimal in this application.
xy-plame
yž-plane
yz-plane
X2- xz-plane
xz-plame
1.0
[mis]
[mvs]
1.0
[mA)]
[nvi]
[nvi]
Simulation
2.5
0.5
-0.5
0.5
10"
[m/4]]
[m/d]
[m/s]
(m/))
MOPINN GD
2.5
y
1.5
y.5
-05
0.0
10"
1.0
(m/)]
[mv]]
1.0 (mva)
[mbg]
1.0
[mye]
10X TDXPDE loss
2.6
y
1.5
y0.5
0.0
0.0
0.0
10'
[mva]
(m/s)
[m/g)]
(mve)
20x PIEE loza
2.5
0.5 y
00.5
2
-05
10"
10X boundary loss
[m/a]]
[mva]]
[mbb
[m/i]]
[nvi]
2.5
3
0.5 y
1.5
-0.
-05
2
0.0
0.0
0.0
0.0
0.5
1.0
0.0
0.5
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Figure A.15: Comparison of a-velocities between CFD simulation, MDPINN-GD, and MDPINN-GD with different loss ratios with absolute errors. Viewed at: ry-plane at 2 0.5 (mid-plane), yz-plane inlet of subdomain yz-plane outlet of subdomain 5, æz-plane inlet of subdomain and æz-plane at outler of subdomain 9.
23

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the
paper's contributions and scope?
Answer: [Yes]
Justification:
The claims made by the authors include: 1) MDPINN-GD significantly
improves fluid prediction results, 2) global continuity loss necessary for accurate prediction results, and 3) the method demonstrated is applicable to systems with large aspect ratios.
These claims have been explored and reflected upon throughout this paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results be expected to generalize other settings.
It is fine include aspirational goals as motivation long is clear that these goals are not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations of the work performed by the authors?
Answer: [Yes]
The main limitations of the paper include: 1) MDPINN-GD z-velocity predic- tion inaccuracies, and 2) demonstrations other geometries. These limitations have been addressed in the conclusion section with limitations subsection.
Justification:
Guidelines:
The answer NA means that the paper has limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create separate "Limitations" section in their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g.. independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution low or images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed not penalize honesty concerning limitations.
3. Theory assumptions and proofs
24

---

Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
Justification: This paper does not contain any theoretical results.
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear in the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Question: Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Answer: [Yes]
Justification: All settings for the simulation and MDPINN G-D network are provided in Section 2 Problem setup under 2.1 andd] 3 These details are necessary for reproducing the
main experimental results. pseudocode also provided in the Appendix A.2 for further clarifications.
Guidelines:
The answer NA means that the paper does not include experiments.
If the paper includes experiments, a No answer to this question will not perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken make their results reproducible verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is a novel architecture, describing the architecture fully might suffice, or if the contribution be necessary to either make it possible for others to replicate the model with the same dataset, or provide access to model. In general. releasing code and data often one good way to accomplish this, but reproducibilityy can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g., in the case of large language model), releasing of a model checkpoint, other means that are appropriate to the research performed.
specific model and empirical evaluation, may
While NeurlPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how reproduce that algorithm. (b) If the contribution is primarily the architecture clearly and fully.
model architecture, the paper should describe
(c) the contribution is a new model (c.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open- open-source dataset or instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome describe the particular way they provide for reproducibiliity In the case of closed-source models, it may be that access to the model is limited in
25

---

some way (e.g., to registered users), but it should be possible for other researchers have some path reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
Answer: [No]
We utilized well-established code for PINN, already available from the original contributors. Our primary contribution lies in adopting multiple PINNs with more robust loss terms. This does not require the sharing of code. Instead, pseudocode is provided in the Appendix A.2 for reproducibility.
Justification:
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for new open-source benchmark).
The instructions should contain exact command and environment needed to run to reproduce the results. See the NeurlPS code and data submission guidelines (https
/ //nips.co... ctaipsscc/publlc/guiicc httpa///gguiieee/eeeeuubbbiiiiia swi.suubiiiiiiooooooooooooo for more details.
The authors should provide instructions on data access and preparation, including how access the raw data, preprocessed data, intermediate data, and generated data,
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. If only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information as possible in supplemental material (appended the paper) is recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: All settings for the simulation and MDPINN-GD network are provided in Section 2.1 and Section 3
Guidelines:
The answer NA means that the paper does include experiments.
The experimental setting should be presented the core the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [Yes]
Justification: This paper presents the mean and max values mean absolute errors. Nor- malization is also done with respect to the inlet velocity to show the significance of the change in error. These are presented in the Section with additional material provided in the Appendix A.5 and Appendix A.7
26

---

Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results are accompanied by error bars, confi- dence intervals, or statistical significance tests, least for the experiments that support the main claims of the paper.
The factors of variability that the enror bars capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall
run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
is OK to report I-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables figures symmetric error bars that would yield results that are out range (c.g. negative error rates).
If error bars reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?
Question:
Answer: [Yes]
Justification:
The paper has indicated the type of GPU, computation time, and memory usage for the experiments. These are provided in Section 4.3
Guidelines:
The answer NA means that the paper does include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).
9. Code of ethics
Question: Does the research conducted the paper conform, in every respect, with the NeurlPS Code of Ethics
Answer: [Yes]
Justification:
This paper conforms to all aspects of the NeurIPS Code of Ethics.
Guidelines:
The answer NA means that the authors have reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special cireumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g., if there is special consid- eration due to laws or regulations in their jurisdiction).
Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
27

---

Answer: [Yes]
Justification: The Section and Section discusses the positive societal impacts of MDPINN- GD. Negative societal impacts of the proposed network architecture are not anticipated as experiments do not involve sensitive data or information and the proposed methodologies complies with law of physics.
Guidelines:
The answer NA means that there is no societal impact of the work performed.
If the authors answer NA or No. they should explain why their work has no societal impact or why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g.. deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).
11. Safeguards
Question:
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Answer: [NA]
Justification:
The work presented in this paper poses no risk to misuse.
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but encourage authors to take this into account and make best faith effort.
12. Licenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: References to the previous PINNs and multi-domain methods properly cited. The use of PyTorch has been properly disclosed. Simulation data used has been generated by the authors, with disclosure made regarding the use of Ansys Fluent.
28

---

Guidelines:
The answer NA means that the paper does not use existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset used and, possible, include URL.
The name of the license (e.g., CC-BY 4.0) should be included for each asset. For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms use in the package should be provided. For popular datasets, pitaasetsspapessiiiii paperswi thcode. com comddatasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. If this information is not available online, the authors are encouraged to reach out to the asset's 's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification: This paper does release new assets. The framework MDPINN-GD is well documented throughout the paper and can be referenced without license.
Guidelines:
The answer NA means that the paper does release new assets. Researchers should communicate the details the dataset/code/model as part their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent was obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given participants and screenshots, if applicable, as well as details about compensation (if any)?
Answer: [NA]
Justification: This research and the corresponding paper does not involve crowdsoureing research with human subjects.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Including this information the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then much detail as possible should be included the main paper.
According to the NeurlPS Code Ethics, workers involved in data collection, curation, other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals equivalent for research with human subjects
Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
29

---

Answer: [NA]
Justification: This research and the corresponding paper does not involve crowdsourcing nor research with human subjects. No IRB approvals were seeked/needed.
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors adhere the NeurIPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such as institution conducting the review.
16. Declaration of LLM usage
Question: Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods this research? Note that the LLM is used only for writing, editing, formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Answer: [NA]
Justification: This research does not involve LLMs any important, original, non- standard components. LLM was only used for writing, editing or formatting purposes.
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs as any important, original, non-standard components.
Please refer to L.LM policy httpa////eerrrps..cc/eeeeeeeeeee for what should or should not be described.
30


Reviews:
[{"review_id": "EcsGINy2Q2", "reviewer": "Reviewer_VWEc", "summary": "The authors in this work study the efficacy of PINNs and specially with domain decomposition techniques in solving a forward problem involving complex steady state three dimensional intersecting flows in non-rectangular domains. They proposed a Multi-Domain PINN (MD-PINN) which decomposed the complex 3D non-rectangular domain into smaller cuboidal domains to which individual PINN subnetworks are assigned to. These subnetworks are trained simultaneously in parallel through a composite loss function that comprises the physics residual loss, boundary conditions loss, interface matching loss and global mass conservation (global dynamics) loss components all with equal contributions. With the addition of the global dynamics constraint / global mass conservation loss the MDPINN-GD variant is obtained. While the interface matching loss imposes zeroth order continuity of field variables at the shared boundaries of the subdomains, the global dynamics constraint ensures that the global mass continuity is satisfied at the inflow and outflow boundaries of the entire domain. The authors demonstrate that MDPINN and specially MDPINN-GD perform better than a single network PINN model for such flow problems.", "questions": "- *Introduction & literature survey:* The literature survery feels incomplete as there are multiple domain decomposition based PINNs that have come out in the recent years not just limited to steady state flows. Consider works such as Conservative-PINNs, XPINNs, DPINNs, DPIELMS, unified-causal framework, etc. **Why have the authors not considered works of Dwivedi et al (DPINNs/ DPIELMS), Jagtap (XPINNs), and more unified work of Penwarden (although for time domain decomposition which has solve much more complex unsteady problems)?** \n- *Problem considered:*  Although the authors claim that these only consider 2D systems, the 3D system considered is only a steady state system and that too at a low reynolds number of 247.5. **Since the authors are already using domain decomposition, why not go for higher reynolds number where stronger gradients might exist and where MDPINN would possibly outshine single PINN even better?**\n- *Methodology: (single PINN)* The authors did not explore recent advancements in PINN training (see Sifan Wang / Aditi Krishnapriyan's works on failure modes and advanced training strategies) for their problem. PINNs can perform fairly well on steady state problems even if multidimensional when appropriately constructed (see Sifan wang's work on *\"An expert's guide to training physics-informed neural networks\"* with some of these strategies highlighted). With very thorough hyperparameter tuning and adopting adaptive sampling/ loss component weighting strategies, a single PINN network should suffice. **Why not carry out loss balancing by first weighing down the physics loss by a factor or two and test how PINN results evolve?** Such a study becomes mandatory to claim SOTA of MDPINN. \n- *Methodology: (loss formulation):* The authors have not used global or local loss component weighting which has always resulted in signficant improvement in the performance of PINN models. Given the steady flow scenario, the performance improvement would be visible and easy to obtain. A weighted summation of loss components would have been best suited in this work even if hand tuned: \n$$\\mathcal{L} = \\lambda_{BC}\\mathcal{L}_{BC} + \\lambda_{Phy}\\mathcal{L}_{Phy} + \\lambda_{M}\\mathcal{L}_M + \\lambda_{V}\\mathcal{L}_{V}$$ Different weighting strategies - globally fixed but physics loss relaxation, adaptive loss weighting using gradnorm - residual adaptive weighting, etc, all of which have been recent advancements in the field. **Atleast one of such methods need to be considered to improve PINN and MDPINN overall.** \n- *Methodology: (activation function, sampling):* The activation function or the sampling methodology for the collocation points are not specified explicitly. Moreover, the sampling strategy is not justified as well. If strong gradients are expected at the boundaries, why not have higher collocation point density over there? **This has larger implications on the training and performance of single network PINN. Tailored sampling, and/or loss weighting all are nowadays default and best practice settings for training PINN models.**\n - *Results: (Error metrics):* Since loss weighting / adaptive sampling have not been tested on single network PINN, it cannot be completely ruled out that single network PINN model is not performant for a steady state problem. Interestingly, PINNs have shown to perform well on arbitrary domain (hidden fluid mechanics - Raissi et al. (2019)). Moreover, this can be explained by the performance metrics as well, where the improvements are less than 100% for MDPINN/ MDPINN-GD. Since the improvement is not really by a significant factor, using loss balancing could significantly improve not just the PINN model performance but MDPINNs could also benefit from such strategies. **It would be advisable to provide sectional velocity and pressure profiles to identify where the mismatch is more pronounced especially in the mixing zones of the flow.** \n- **Compuational cost** Why have the authors not highlighted the computational cost of training the single PINN and MDPINN/ MDPINN-GD models? This information is crucial when compared with typical CFD simulations. Both training and inference times have to be provided in a table in comparison to CFD simulations for reference. PINNs are usually slow to train but quick to infer at any given location and any discretisation level. **So any computational efficiency claims have to be justified with such a table mentioned above.**\n \nOverall, the method although not completely novel, the application is original. While it is agreeable that domain decomposition is what improves MDPINN performance, it does not rule out the possibility of single PINN model performing equally well when SOTA training strategies such as loss balancing/ adaptive collocation point sampling are adopted.", "limitations": "Yes. The authors have been diligent in providing the limitations of their work which they plan to address in future work. \nHowever, here is a suggestion to further improve the quality of the paper. Thorough evaluation of loss weighting for PINN is a necessary test to rule out the possibility of single network PINN performing well for such complex 3D problems. Hence, it is suggested that authors comment on this or carry out further analysis using loss weighting for single PINN models and provide the error metrics for the same.", "rating": 5, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The key strengths and weaknesses of the paper are listed below. It is requested that the authors address them. \n\n**Strengths:**\n  - The authors chose a complex 3D domain and detailed the problem formulation very briefly but clearly explaining the choice of their design domain, governing equations.\n  - The loss formulation, multi-domain decompostion and global dynamics constraint are clearly developed and detailed without any confusion. This aids in understanding transparently what is done by the authors and what is left out. \n  - The authors also highlight potential limitations to be addressed through extended studies in the appendices.\n  - The authors have only used mean and maximum absolute errors as the evaluation metrics but have carried out a comprehensive evaluation over each subdomain, and variable (3 dimensional velocity components and pressure)\n  - The quality and clarity in terms of presentation of the idea, method and results are good. \n\n**Weaknesses:**\n - *Introduction & literature survey:* While the motivation to use multidomain deomposition for 3D problems is clear in the Introduction, the literature survery feels incomplete as there are multiple domain decomposition based PINNs that have come out in the recent years not just limited to steady state flows. Consider works such as Conservative-PINNs, XPINNs, DPINNs, DPIELMS, unified-causal framework, etc.\n- *Problem considered:*  Although the authors claim that these only consider 2D systems, the 3D system considered is only a steady state system. However, the problem considered is original.\n- *Methodology: (single PINN)* The authors did not explore recent advancements in PINN training (see Sifan Wang / Aditi Krishnapriyan's works on failure modes and advanced training strategies) for their problem. PINNs can perform fairly well on steady state problems even if multidimensional when appropriately constructed (see Sifan wang's work on *\"An expert's guide to training physics-informed neural networks\"* with some of these strategies highlighted). With very thorough hyperparameter tuning and adopting adaptive sampling/ loss component weighting strategies, a single PINN network should suffice. Such exploration has not been done\n- *Methodology: (loss formulation):* The authors have not used global or local loss component weighting which has always resulted in signficant improvement in the performance of PINN models. Given the steady flow scenario, the performance improvement would be visible and easy to obtain. A weighted summation of loss components would have been best suited in this work even if hand tuned: \n$$\\mathcal{L} = \\lambda_{BC}\\mathcal{L}_{BC} + \\lambda_{Phy}\\mathcal{L}_{Phy} + \\lambda_{M}\\mathcal{L}_M + \\lambda_{V}\\mathcal{L}_{V}$$\n- *Methodology: (activation function, sampling):* The activation function or the sampling methodology for the collocation points are not specified explicitly. Moreover, the sampling strategy is not justified as well. This has larger implications on the training and performance of single network PINN. \n - *Results: (Error metrics):* SInce loss weighting / adaptive sampling have not been tested on single network PINN, it cannot be completely ruled out that single network PINN model is not performant for a steady state problem. Interestingly, PINNs have shown to perform well on arbitrary domain (hidden fluid mechanics - Raissi et al. (2019)). Moreover, this can be explained by the performance metrics as well, where the improvements are less than 100% for MDPINN/ MDPINN-GD. Since the improvement is not really by a significant factor, using loss balancing could significantly improve not just the PINN model performance but MDPINNs could also benefit from such strategies.", "was_revised": true, "final_justification": "I have improved my scores based on the corrections and improvements made to the manuscript. Especially, the clarity of the document has now improved with respect to clarity and significance owing to additional comparisons in the appendix, and in the main text highlighting the background better.", "rebuttal": "We sincerely appreciate the reviewers' thoughtful and constructive feedback, which will significantly enhance the clarity and strength of our manuscript. We will carefully address each comment in our revised paper and future work.\n\n1. Introduction & literature survey: We appreciate the suggestion to strengthen the literature survey. Further explanations will strengthen the motivation of our proposed work. Here, we will briefly share the summary in the publication order and will include them in the introduction of the revised manuscript.\n\n* DPINN introduced distributed PINN for each subdomain with additional flux conditions at cell interfaces as local regularizer when combining together. It demonstrates higher accuracy and better data efficiency over PINN. However, the experiments are limited to 2D simple flows.\n* DPIELMS is a hybrid network of PINN and extreme learning machine (ELM), a rapid version with distributed mechanism tackling both stationary and time-dependent problems. Proposed architecture is experimented on 1D, 2D problems.\n* cPINN studied discrete PINN for each subdomain that stitches back through the conservation of flux at the interfaces similar to DPINN. Additionally, an average solution is enforced at the common interfaces. The work highlights the potential of employing different neural network architecture for each subdomain which can be our future direction. Similarly, they only tackled 2D problems.\n* XPINN proposed generalized PINN domain decomposition in both space and time for arbitrary complex geometry domains and any type of PDEs. It applies simple interface conditions to irregular shapes, useful in dynamic problems with moving interfaces. The work is demonstrated on 1D to 3D domains with both time-dependent and independent problems, yet doesn’t involve complex fluid dynamics such as flow mixing.\n* Unified-causal framework by Penwarden et al. is a stacked-decomposition method that investigates causes of training challenges and relates to information propagation, tackling time-dependent PDEs with no data. It also leverages transfer learning to initialize subnetworks and loss toleranced-based propagation for the subdomains, resulting in computational speedup. The work tackles complex transient problem yet limited to the 1D domain. The concept of information propagation can be useful for spatial domain if the subdomains are partitioned from inlet to outlet, which can be a meaningful future research direction.\n\n2. Methodology (single PINN): We appreciate directing our attention to recent advancements in PINNs. In our study, which focuses on 3D fluid dynamics involving complex flow structures and geometries (i.e., flow mixing with multi-inlet, multi-outlet channel), we believe that single-network PINN approach faces practical challenge. Our literature review indicates that successful applications of PINNs are largely limited to simplified 3D configurations, such as low-aspect-ratio channels and single inlet/outlet. These cases do not fully capture the complexities of 3D systems with intricate flow structures (even in laminar flow).\nWhile single-network PINNs can also be trained for such problems, their optimization requires meticulous tuning. In contrast, MDPINN offer a robust and efficient alternative, which achieves improved accuracy with the exact same architecture as single-network PINN. By distributing computational burden across multiple domains, it mitigates the training challenges, such as numerical stiffness and poor convergence due to highly nonlinear physics. Details have been omitted in the paper due to the focus, but we have tried a variety of hyperparameter tuning and loss weighting strategies, which didn’t lead to significant improvement. In the revised manuscript, we will include selected experimental results on loss balancing in the appendix.\nTo our knowledge, this work is the first to demonstrate successful application of MDPINN to a 3D cross-flow configuration, achieving accurate solutions without the need for extensive hyperparameter tuning or adaptive strategies. We will include the literature survey on Sifan’s and Krishnapriyan’s work in the revised manuscript.\n\n3. Methodology (loss formulation): Thank you for your valuable feedback. In Section 3.3, we introduce a composite loss function for our MDPINNs. This function does not include loss weights, but we conducted experiments with different weight ratios of the loss components based on Sifan’s work below. A few examples of ratios we explored between boundary and PDE losses include: 1:5, 1:10, and 1:50. We observed well-matching predictions on boundary points, hence, we increased the PDE loss weight, however, it didn’t lead to noticeable performance improvement over equal weights. We acknowledge that optimizing loss component weights could enhance PINN performance and will experiment on various combinations. To clarify further, we will include a few experimental results on loss weight balancing in the appendix of the revised manuscript.\n\nWang, Sifan, Yujun Teng, and Paris Perdikaris. \"Understanding and mitigating gradient flow pathologies in physics-informed neural networks.\" SIAM J. Sci. Comput. 43.5, A3055-A3081 (2021).\n\n4. Methodology (activation function, sampling): We acknowledge the suggestion regarding tailored sampling to increase collocation point density in regions with strong gradients. While our current approach with MDPINNs and Latin hypercube sampling achieved satisfactory performance, we agree that adaptive sampling strategies could further enhance accuracy. We experimented with adaptive sampling based on the loss magnitude, in which the top 10% of the highest magnitude collocation points are sampled as additional collocation points (corners and boundaries). This method follows a similar approach provided by the sources below. While we observed minimal improvement on limited cases, this is a valuable direction for future work for further optimization.\n\tThe activation function and sampling methodology for our MDPINN and single-network PINN are detailed in Section 3 and 4.2, respectively. We employed adaptive sine activation function to enhance the model's ability to capture complex physics. We used Latin hypercube sampling to generate 10,000-40,000 collocation points randomly distributed in each subdomain.\n\nC. Wu et al. \"A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks.\" Comput. Methods in Appl. Mech. and Eng. 403, 115671 (2023).\n\nZ. Mao and M. Xuhui. \"Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions.\" Appl. Math. Mech. 44.7, 1069-1084 (2023).\n\n5. Results (Error metrics): Thank you for insightful feedback. In our work, we aim to demonstrate that MDPINNs simplify training while achieving robust performance compared to PINNs under similar training conditions, achieving 75% accuracy improvement, which is considered relatively high among existing works, though not exceeding 100%. We acknowledge that incorporating loss weighting or adaptive sampling could further enhance the performance, which we consider as a valuable direction for future work.\nAdditionally, we note that Raissi et al. (2019) addresses significantly simpler fluid mechanics problems, specifically 2D flows, low geometric aspect ratio and no cross-flow structures. Furthermore, it leverages concentration field data to reconstruct velocity fields, whereas our work focuses on a purely data-less prediction. This distinction highlights the increased complexity of our problem setup, where MDPINNs demonstrate sufficient success compared to PINNs.\n\n6. Problem considered: We appreciate raising an important question of applying our MDPINNs to higher Reynolds number flows. Our study deliberately focused on a steady-state 3D system at a Reynolds number of 247.5 as a foundational step in applying PINNs to complex 3D fluid dynamics problems. Despite laminar flow, the chosen cross-flow configuration introduces intricate 3D mixing, deflection and flow redistribution at the intersection. To the best of our knowledge, no prior work has successfully applied PINNs to such 3D configuration. This represents a non-trivial challenge, as the interplay of complex geometries and flow structures tests the limits of state-of-the-art PINN methodologies.\nThe decision to focus on laminar flow governed by the 3D steady-state Navier-Stokes equations allowed us to establish a robust baseline for MDPINNs in a controlled yet challenging setting. As the reviewer notes, higher Reynolds number flows (Re > 1000) introduce unsteady and turbulent characteristics, necessitating the use of more complex governing equations, such as Reynolds-Averaged Navier-Stokes or other turbulence models. These require not only significant modifications to the PINN framework but also extensive hyperparameter tuning and potentially novel adaptive strategies to handle the increased nonlinearity and numerical stiffness. Such complexities are beyond the scope of our current work and we will investigate in future work. \n\n7. Results (Error metrics): Sectional views of velocity and pressure profiles for all the models at 1) center subdomains where flow mixing occurs and 2) outlet subdomains, are shown in the Appendix A.3 and A.6 for further comparisons.\n\n8. Computational cost: The training cost of MDPINN model is discussed in Section 4.3. We agree that a thorough comparison of inference efficiency between PINN-based methods and CFD simulations is critical for evaluating practical benefits. While this work focuses primarily on training dynamics and accuracy improvements, we acknowledge that a systematic benchmarking of inference speed across various discretization levels, is necessary to justify computational efficiency claims. We consider this an important direction for future work and will include a discussion in the revised manuscript."}, {"review_id": "vWBqjpfsMB", "reviewer": "Reviewer_zyPW", "summary": "The paper focuses on s apecial benchamark example CFD problem with PINN method. The computation domain is divided into blocks to improve the efficiency. The results show the improvement of the single PINN method.", "questions": "1. the error seem to be not small (about 0.1), is that a so turblent case\n2. the dimensionaless is great, so are there any plan to apply it for the real scale problem\n3. how deos the inlet/outlet condition influence the problem or convergence?", "limitations": "N/A", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Pros:\n1. the point of view for the paper is great. It is practical for the problem with the tune and the block-wise implementation actually provide some insight to divide and conquer the problem.\n2. the part described for the domain design is great I believe, many paper (even accepted) not focus on the CFD details (though I think I should in the appendix....)\n\nCons:\n1. the paper is not so friendly for readers. The results are written most like a report. The illusion might not point out the key contribution\n2. the blocked-wise design and the transmisson on the block interface is not non-trivial. More complex and large geometry should be provided to better characteristize the advantages of the proposed methods.\n3. How about comparison with non-PINN methods, like classic simulation since the block-wise can be viewed as a kind of discretization.\n4. the training time is still long and the cost for GPUs are large\n\nI think the paper is more related to the computational physics but not mainly the learning as the main contribution. But I still give positive since it tries to design for a real CFD problem.", "was_revised": true, "final_justification": "I believe my concerns about the complexity of the geometry, computer cost, and some possible comparisons with classic methods remain unresolved. The author tends to state them as future work. I will keep my score since I think some pts like the dimensionless analysis and more settings and methods from CFD make the paper different from the ML papers. These pts are positive for the AI4CFD field. So I just keep my score.", "rebuttal": "We sincerely appreciate the reviewers' thoughtful and constructive feedback, which will significantly enhance the clarity and strength of our manuscript. We will carefully address each comment in our revised paper and future work.\n\n1. Comment: The paper is not so friendly for readers. The results are written most like a report. The illusion might not point out the key contribution: \n\nResponse: We thank the reviewer for this feedback about the paper style. We have edited the style to be more aligned with similar NeurIPS publications and highlighted main contributions and lessons learned from this research, which will be more reader-friendly in the revised manuscript.\n\n2. Comment: The blocked-wise design and the transmission on the block interface is not non-trivial. More complex and large geometry should be provided to better characterize the advantages of the proposed methods: \n\nResponse: We appreciate the reviewer for thoughtful comments regarding the complexity of the design. While the concept of blocked design and the transmission at interfaces may seem trivial in some contexts, their effective application to 3D fluid dynamics problems represents a significant contribution, particularly given the computational challenges associated with modeling complex flow structures and geometries. Our work demonstrates the successful implementation of PINNs in addressing such problems for the first time, intricate configurations like multi-inlet, multi-outlet channel flow mixing. As demonstrated in the paper, the single-network PINN approach faces practical challenges that warrant the exploration of alternative strategies, especially regarding the training without any data. Our literature review indicates that existing applications of PINNs in fluid dynamics are largely limited to simplified 3D configurations, such as low-aspect-ratio rectangular channels and/or low Reynolds numbers. These cases do not fully capture the complexities of 3D systems with intricate flow structures arising from complex geometries, such as the cross-flow configuration addressed in our work. As this approach enhances the scalability, we agree that extending the work to more complex and large geometry will be valuable, which we will address in the future work. \n\n3. Comment: How about comparison with non-PINN methods, like classic simulation since the block-wise can be viewed as a kind of discretization: \n\nResponse: We thank the reviewer for their valuable feedback and raising question about comparisons with non-PINN methods, specifically referencing “classic simulation” in the context of discretization. The reviewer’s keyword of “classic simulation” is somewhat ambiguous, as our paper already includes a comparison with a computational fluid dynamics (CFD) simulation based on finite volume methods (FVM), which we consider a standard non-PINN method employing discretization. As detailed in the paper, Figures 3 and 4 present a direct comparison between the flow field predictions from our MDPINNs and those from a CFD simulation based on traditional discretization techniques. Specifically, Figure 4 illustrates the absolute error distribution, demonstrating that the mean absolute velocity errors of MDPINNs are below 14.9% compared to the CFD benchmark. This comparison highlights the competitive performance of MDPINNs relative to established numerical methods. If the reviewer had a specific classic simulation method in mind besides CFD, we would appreciate further clarification to tailor our response or future comparisons more precisely.\n\n4. Comment: The training time is still long and the cost for GPUs are large: \n\nResponse: We appreciate sharing the comments on the cost of the proposed work. We acknowledge that the training time is long and GPU usage is high. Similar to other scientific ML work, a motivating factor of this research is the emphasis on application of PINNs for iterative design processes. CFD requires rerunning of the entire simulation on a different design which incurs linear computational runtime with the number of designs to evaluate. On the other hand, in regards to ML-driven models, once the model is trained, the inference on new design is rapid, which can be considered closely to constant time evaluation. In practical design optimization process, thousands or even more than ten thousands of designs can be compared. In these cases, PINNs have superior computational performance compared to traditional methods and can overcome the expensive training time. For the memory consumption in GPU, it is mainly coming from extensive derivatives in PDE computation. There are a several ways to improve this, such as 1) sharing the same variables of subdomain PDE residuals/boundary values for the global conservation, 2) removal of unused 2nd order derivatives (e.g., d3F/dxdydz) that are automatically computed by using 3D tensors, to be replaced with individual 1D tensors. These can be crucial enhancements to MDPINN, but are considered outside of the proposed scope. We will apply these updates in future work for better practicality.\n\n5. Comment: The error seem to be not small (about 0.1), is that also in turbulent case: \n\nResponse: We thank you for sharing the comments on the experimental results and inquiring about turbulent cases. We agree that there is yet some discrepancy between the predicted solutions (MDPINN) and the ground truth (CFD). However, we have significantly improved the accuracy (53-75%) on a 3D complex problem from the original PINN, which is very widely used with 15000+ citations. We demonstrated the methodology to 3D fluid dynamics problems involving complex flow structures and geometries (i.e., flow mixing with multi-inlet, multi-outlet channel).\nRegarding the flow, our study deliberately focused on a steady-state 3D system at a Reynolds number of 247.5 (laminar) as a foundational step. While the Reynolds number is relatively low, the chosen cross-flow configuration introduces significant complexity through intricate 3D mixing, deflection, and flow redistribution at the intersection region. To the best of our knowledge, no prior work has successfully applied PINNs—whether single-network or multi-network—to such a 3D configuration, even at low Reynolds numbers. This represents a non-trivial challenge, as the interplay of complex geometries and flow structures tests the limits of state-of-the-art PINN methodologies. The decision to focus on a steady-state, laminar flow governed by the 3D steady-state Navier-Stokes equations allowed us to establish a robust baseline for MDPINNs in a controlled yet challenging setting. Higher Reynolds number flows (turbulence) introduce unsteady and turbulent characteristics, necessitating the use of more complex governing equations, such as Reynolds-Averaged Navier-Stokes (RANS) or other turbulence models. These require not only significant modifications to the PINN framework but also extensive hyperparameter tuning and potentially novel adaptive strategies to handle the increased nonlinearity and numerical stiffness. Such complexities, while important, are beyond the scope of our current work. We will study the impact of MDPINN on turbulent problems in future work.\n\n6. Comment: The dimensionless is great, so are there any plan to apply it for the real scale problem:  \n\nResponse: Thank you for your positive feedback on our use of dimensionless analysis. In response to your question about applying our approach to real-scale problems, we confirm that our future work includes plans to extend our MDPINN to larger-scale applications. Additionally, we plan to explore advanced optimization algorithms, such as adaptive learning rate methods and gradient compression, as well as model pruning and quantization, to accelerate the training of multiple artificial neural networks simultaneously. These improvements will enable us to tackle real-scale problems effectively while maintaining computational feasibility.\n\n7. Comment: How does the inlet/outlet condition influence the problem or convergence: \n\nResponse: We appreciate your insightful question regarding the influence of inlet and outlet conditions on the problem and its convergence. Inlet and outlet boundary conditions play a pivotal role in determining the complexity of all fluid dynamics problems. Specifically, the inlet condition significantly affects the flow regime: high-velocity inflows can lead to turbulent and unsteady flow, necessitating advanced computational approaches to capture the associated complexities, while low-velocity inflows typically result in laminar and steady flow, allowing for simpler partial differential equations (PDEs) to be solved. However, we increased the complexity by introducing multiple inlets and multiple outlets to induce complex flow mixing. In this work, we focused on a low flow rate scenario, which induced laminar and steady flow, enabling us to address a more tractable set of PDEs. This foundational study establishes a robust framework for our future work, where we plan to extend our approach to high flow rate conditions involving turbulence.\nRegarding outlet conditions, we adopted the commonly used zero-pressure condition, which assumes the flow exits the system without additional constraints. While outlet conditions can influence the solution, their impact is generally less significant than that of inlet conditions in most fluid dynamics studies. By carefully defining these boundary conditions, we ensured stable convergence in our computations, laying the groundwork for addressing more complex scenarios in subsequent research."}, {"review_id": "soOlhgdVuz", "reviewer": "Reviewer_xQ6y", "summary": "The paper introduces a multi-domain physics-informed neural network (MDPINN) framework designed to model complex three-dimensional fluid flows within intersecting channels characterized by high aspect ratios. The key contributions include the implementation of domain decomposition, where the overall domain is partitioned into multiple subdomains, each modeled by an individual PINN, and the enforcement of global mass conservation constraints (MDPINN-GD) to ensure physical consistency across the domain. The framework aims to overcome the scalability and accuracy limitations of traditional PINNs when applied to complex multi-scale 3D geometries. Experimental results demonstrate significant improvements in predictive accuracy.", "questions": "Please see the weakness above.", "limitations": "Yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n1. By employing domain decomposition and enforcing global constraints such as mass conservation, the framework effectively models intricate flow features\n2. The experimental results show substantial improvements, extending existing PINN methodologies to complex 3D geometries, demonstrating its applicability in this broader context. \n\nWeakness\nHowever, overall, this work primarily represents a direct extension of existing methods [21] to three-dimensional problems, without introducing substantial novel algorithmic developments or technical innovations. Specifically, the approach largely involves applying known techniques—such as domain decomposition and global conservation enforcement—to a new problem setting, rather than developing new strategies to address the unique challenges posed by high-dimensional or highly complex geometries. The results mainly serve to validate the applicability of established methods rather than advance the state-of-the-art on fundamental algorithmic or theoretical fronts. \n\n[21] Khemraj Shukla, Ameya D. Jagtap, and George Em Karniadakis. Parallel physics-informed neural networks via domain decomposition. Journal of Computational Physics, 447:110683, December 2021.\n\nIn addition, the manuscript treats the CFD simulation results as ground truth when evaluating the accuracy of the proposed method. This raises the question of whether the manuscript implicitly acknowledges that the method is less accurate than standard CFD solvers. If so, it would be important to clarify whether the proposed approach offers advantages in terms of computational efficiency. More precisely, the manuscript need conduct a fair comparison by adjusting the parameters of the CFD solver to match the accuracy of their method. In such a setting, does the proposed algorithm demonstrate superior efficiency compared to CFD simulations?", "was_revised": true, "final_justification": "After the author's detailed reply, considering discussions with other reviewers and AC, most of my questions have been solved, and thus I have raised the score appropriately.\n\nWhile the initial results show that the method has advantages compared to the NN-based solver, the results are preliminary. I understand it is not possible to perform extensive experiments in the rebuttal phase. But I still think that the paper would benefit from revisions and comparisons of computational cost. In addition, considering the comments of other reviewers, I also think it might be more suitable for the computation physics journal.", "rebuttal": "We sincerely appreciate the reviewers' thoughtful and constructive feedback, which will significantly enhance the clarity and strength of our manuscript. We will carefully address each comment in our revised paper.\n\n1. Comment: However, overall, this work primarily represents a direct extension of existing methods [21] to three-dimensional problems, without introducing substantial novel algorithmic developments or technical innovations. Specifically, the approach largely involves applying known techniques—such as domain decomposition and global conservation enforcement—to a new problem setting, rather than developing new strategies to address the unique challenges posed by high-dimensional or highly complex geometries. The results mainly serve to validate the applicability of established methods rather than advance the state-of-the-art on fundamental algorithmic or theoretical fronts: \n\nResponse: We sincerely appreciate the reviewer’s thoughtful comments and the opportunity to clarify the novelty and contributions of our work. While we acknowledge that our approach builds upon existing PINNs combined with domain decomposition, as referenced in [21], we respectfully disagree with the assessment that our work is merely a direct extension without substantial novel contributions. The reviewer correctly notes that our methodology leverages established techniques, such as domain decomposition and PINNs. However, the experimental results yet showed limited prediction in PINN with domain decomposition, which is much enhanced with a key innovation in our work, global mass conservation enforcement across the entire computational domain, which, to the best of our knowledge, has not been previously investigated in the context of multi-network PINNs (MDPINNs). Unlike the interface-matching boundary conditions described in [21], which focus on ensuring continuity across neighboring subdomains, our approach incorporates global mass conservation as a fundamental constraint in addition to the interface-matching boundary conditions. This is a critical advancement for ensuring physically consistent solutions in complex fluid dynamics problems, particularly in 3D settings, since our work found that only the interface-matching boundary condition is not sufficient to solve such problems. Furthermore, while the reviewer suggests that our work primarily validates the applicability of known methods to 3D problems, we emphasize that there is a significant gap in the literature regarding the application of MDPINNs to 3D fluid mechanics problems with meaningful and practical geometric complexity. Existing demonstrations of MDPINNs in 3D fluid flow, as noted by the reviewer, are limited to overly simplistic geometries, which do not capture the challenges of complex flow structures encountered in practical applications. Our work addresses this gap by adapting MDPINNs to handle 3D fluid dynamics with intricate geometries, requiring novel strategies to integrate global mass conservation and robust benchmarking against computational fluid dynamics (CFD) simulations. This adaptation is far from trivial, as it involves addressing the unique numerical and computational challenges posed by nonlinear physics in high-dimensional, complex domains. We acknowledge that the foundational concept of MDPINNs is not our original contribution, as it builds on prior work. However, our study makes significant strides in advancing the field by demonstrating, for the first time, how MDPINNs can be effectively adapted for 3D fluid dynamics problems with complex flow structures (e.g., flow mixing in multi-inlet, multi-outlet cross-sectional channel).\n\n2. Comment: In addition, the manuscript treats the CFD simulation results as ground truth when evaluating the accuracy of the proposed method. This raises the question of whether the manuscript implicitly acknowledges that the method is less accurate than standard CFD solvers. If so, it would be important to clarify whether the proposed approach offers advantages in terms of computational efficiency. More precisely, the manuscript need conduct a fair comparison by adjusting the parameters of the CFD solver to match the accuracy of their method. In such a setting, does the proposed algorithm demonstrate superior efficiency compared to CFD simulations: \n\nResponse: We thank the reviewer for the comment on our CFD ground truth. CFD has been widely used for several decades as an accurate method to model physical systems, hence, for this research, we presume that the CFD results are what the PINN should converge to, since both processes are governed by identical physical processes without variable influences (i.e., noise, uncertainties). However, in other applications such as the use of experimental data, the ground truth will be the physical measurement dataset, which requires sanitizing. We acknowledge that in this setting, the PINN performance is less than CFD. However, the goal of this research is to demonstrate that certain techniques (domain decomposition, system continuity constraints) are able to vastly improve the network predictions and yield acceptable results (within 14.9% error) for engineering purposes. The CFD settings are tuned independently from the PINN, both tuning processes involved checking for result convergence. We find that in general, CFD requires significantly more nodal points to achieve convergence, approximately 35 times in this specific example. Lastly, a motivating factor of this research is the emphasis on application of PINNs for iterative design processes. CFD requires rerunning of the entire simulation on a different design which incurs linear computational runtime with the number of designs to evaluate. On the other hand, in regards to ML-driven models, specifically PINN/MDPINN in this work, once the model is trained, the inference on new design is rapid, which can be considered closely to constant time evaluation. In these cases, PINNs have superior computational performance compared to traditional methods."}, {"review_id": "XwkjgYsI0W", "reviewer": "Reviewer_A7Dr", "summary": "This paper introduces a decomposition framework along with a global enforcement technique to address the challenges of solving the 3D Navier–Stokes equations in complex, non-canonical geometries.", "questions": "It would be helpful if the authors could clarify whether comparisons have been conducted with traditional solvers or more recent PINN variants. Without such baselines, it is difficult to assess the practical significance of the proposed method.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n\n1. The paper is overall well-written and well-organized.\n\n2. It addresses an important and timely question in scientific computing, and the authors propose a novel regularization approach to enhance physical consistency of the learned models.\n\n3. The benchmark case proposed in this paper appears to be novel and, to the best of my knowledge, has not been presented in previous machine learning conferences.\n\nWeaknesses:\n\n1. The simulation results of the proposed method do not appear to be particularly strong—there is a noticeable discrepancy between the predicted solutions and the ground truth. It remains unclear whether the method’s accuracy is sufficient for real-world applications.\n\n2. The paper lacks a thorough comparison with other advanced PINNs. For example, while some recent works also explore 3D spatial decomposition (e.g., https://pubs.aip.org/aip/pof/article/36/2/021914/3267717), the authors only provide a high-level mention, without any in-depth comparison in terms of methodology or numerical performance.\n\n3. The paper does not compare its results with traditional numerical solvers. Since training PINNs is often computationally expensive, it is important to benchmark both efficiency and accuracy against traditional solvers such as FDM or FVM, to justify the practical advantages of the proposed approach.", "was_revised": true, "final_justification": "While the style of this submission may differ from typical NeurIPS papers, I appreciate the cross-disciplinary perspective and the potential impact on scientific computing and engineering applications.", "rebuttal": "We sincerely appreciate the reviewers' thoughtful and constructive feedback, which will significantly enhance the clarity and strength of our manuscript. We will carefully address each comment in our revised paper.\n\n1. Comment: The simulation results of the proposed method do not appear to be particularly strong—there is a noticeable discrepancy between the predicted solutions and the ground truth. It remains unclear whether the method’s accuracy is sufficient for real-world applications: \n\nResponse: We appreciate the reviewer’s thoughtful comments and the opportunity to clarify the novelty and contributions of our work. We agree that there is yet some discrepancy between the predicted solutions (MDPINN) and the ground truth (CFD). However, as emphasized in the introduction, our work targets the engineering design community’s need for computationally efficient methods to evaluate fluid dynamics systems. Unlike traditional CFD, which incurs linear computational runtime for each design iteration, our PINN/MDPINN approach enables rapid inference on new designs with near-constant time evaluation after training. In practical design optimization, where thousands of designs may be compared, this efficiency is critical, even if it involves some trade-off in accuracy. We will clarify this balance between computational speed and predictive accuracy for real-world applications in the revised manuscript.\n\tMoreover, we have significantly improved the accuracy (53-75%) on a 3D complex problem from the original PINN, which is very widely used with 15000+ citations. We demonstrated the methodology to 3D fluid dynamics problems involving complex flow structures and geometries (i.e., flow mixing with multi-inlet, multi-outlet channel) that closely mimics the complexity of real-world applications. As demonstrated in the paper, the single-network PINN approach faces practical challenges that warrant the exploration of alternative strategies, especially regarding the training without any data (either from CFD simulation or experimental measurements). Our literature review indicates that existing applications of PINNs in fluid dynamics are largely limited to simplified 3D configurations, such as low-aspect-ratio rectangular channels and/or low Reynolds numbers. These cases do not fully capture the complexities of 3D systems with intricate flow structures arising from complex geometries, such as the cross-flow configuration addressed in our work. To our knowledge, this work is the first in the fluid dynamics field to demonstrate the successful application of multi-domain PINNs to a 3D cross-flow configuration.\n\n2. Comment: The paper lacks a thorough comparison with other advanced PINNs. For example, while some recent works also explore 3D spatial decomposition (e.g., https://pubs.aip.org/aip/pof/article/36/2/021914/3267717), the authors only provide a high-level mention, without any in-depth comparison in terms of methodology or numerical performance: \n\nResponse: We sincerely appreciate the reviewer’s thoughtful comments and the opportunity to clarify the novelty and contributions of our work. Gu’s work (2024) referenced below, has studied domain decomposition PINN method for solving incompressible Navier-Stokes problem. They investigated various continuity conditions at interfaces to analyze the performance on the predictive accuracy. Additionally, they explored 1) dynamic weight method, 2) attention mechanism to mitigate gradient pathologies. However, their experiments are limited to 2D steady flow, 3D unsteady flow and single-inlet, two-outlet design problems. \nOur work extends the problem to 3D multi-inlet, multi-outlet problem, introducing flow mixing, a complex, intricate flow dynamics. On top of the interface-matching boundary conditions described in Gu’s work, which focus on ensuring continuity across neighboring subdomains, our approach incorporates global mass conservation as a fundamental constraint in addition to the interface-matching boundary conditions. This is a critical advancement for ensuring physically consistent solutions in complex fluid dynamics problems, particularly in 3D settings, since our work found that only the interface-matching boundary condition is not sufficient to solve such problems. We also explored the dynamic weight method on the loss terms, which are omitted in the paper as no significant improvement was observed as the dominating contribution was obtained via global dynamic conservation.\nWe believe the literature survey on the suggested paper can be further clarified and will include them in the revised manuscript.\n\nGu, Linyan and Qin, Shanlin and Xu, Lei and Chen, Rongliang. \"Physics-informed neural networks with domain decomposition for the incompressible Navier-Stokes equations.\" Physics of Fluids 36, 021914 (2024).\n\n3. Comment: The paper does not compare its results with traditional numerical solvers. Since training PINNs is often computationally expensive, it is important to benchmark both efficiency and accuracy against traditional solvers such as FDM or FVM, to justify the practical advantages of the proposed approach: \n\nResponse: We thank the reviewer for providing this comment, and for raising the question about comparisons with traditional numerical solvers, specifically FDM or FVM. Our paper already includes a comparison with a computational fluid dynamics (CFD) simulation based on finite volume methods (FVM), which we consider a standard numerical solver employing discretization. As detailed in the paper, Figures 3 and 4 present a direct comparison between the flow field predictions from our MDPINNs and those from a CFD simulation based on traditional discretization techniques. Specifically, Figure 4 illustrates the absolute error distribution, demonstrating that the mean absolute velocity errors of MDPINNs are below 14.9% compared to the CFD benchmark. This comparison highlights the competitive performance of MDPINNs relative to established numerical methods. \nRegarding the training time of the MDPINN-GD model, Section 4.3 has the listed details. Similar to other scientific ML work, a motivating factor of this research is the emphasis on application of PINNs for iterative design processes. CFD requires rerunning of the entire simulation on a different design which incurs linear computational runtime with the number of designs to evaluate. On the other hand, in regards to ML-driven models, specifically PINN/MDPINN in this work, once the model is trained, the inference on new design is rapid, which can be considered closely to constant time evaluation. In these cases, PINNs have superior computational performance compared to traditional methods. In the revised manuscript, we will include thorough comparison of simulation runtime and inference efficiency between PINN-based methods and traditional CFD simulations, which is helpful for evaluating practical benefits."}]

Author rebuttal:

Meta-review: This paper proposes a multi-domain decomposition physics-informed neural network (PINN) designed to handle 3D non-rectangular domains governed by nonlinear fluid dynamics. The primary methodological contribution is the introduction of a **global mass conservation loss term**, complementing existing interface matching terms. Experimental results demonstrate encouraging performance compared with numerical CFD baselines.

**Strengths:**

* Tackles a challenging and practically relevant application: 3D complex fluid dynamics, which is rarely addressed in the PINN literature.
* Introduces a principled modification --- global mass conservation loss term---  that enhances the physical fidelity of the model.
* Addresses an application scenario that could broaden the impact of PINNs within the NeurIPS community, moving beyond simplified or toy problems.

**Concerns:**

* The experimental validation is relatively limited. Reviewers expect additional results, including:

  * Comparisons with more advanced PINN and multi-domain PINN approaches.
  * Ablation studies on the role of different loss ratios.
  * Sensitivity analysis with respect to CFD mesh resolution/granularity.
  * More systematic benchmarking of inference speed to substantiate the claimed advantages (slower training but faster transfer/adaptation).
* A more thorough discussion of related work in multi-domain PINNs would strengthen the paper. For example, Li et al. (ICML 2023) show that interface matching significantly affects performance and propose a meta-learning approach to optimize interface conditions.

**Assessment:**

Despite the above concerns, the reviewers agree that the paper makes a meaningful step toward addressing **realistic, high-dimensional, and non-trivial PDE problems** with PINNs. This direction is timely and important for the community.

**Recommendation:**

I recommend **acceptance**. However, I strongly encourage the authors to follow through on their commitment (as noted in the rebuttal) to strengthen the experimental results in the camera-ready version. This will greatly improve the paper’s clarity, rigor, and impact.

**Reference:**
\[1] Shibo Li, Michael Penwarden, Yiming Xu, Conor Tillinghast, Akil Narayan, Robert Kirby, and Shandian Zhe. *Meta Learning of Interface Conditions for Multi-Domain Physics-Informed Neural Networks*. ICML 2023.
Decision: Accept (poster)