Title: Physics-informed machine learning with domain decomposition and global dynamics for three-dimensional intersecting flows
Venue: NeurIPS 2025
Accepted: yes
Keywords: Physics-informed neural networks

domain decomposition

global dynamics

three-dimensional fluid dynamics
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.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\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.\n\nQUESTIONS:\nIt 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.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\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?\n\nQUESTIONS:\nPlease see the weakness above.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-07-02"}

comments: 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.

STRENGTHS AND WEAKNESSES:
Pros:
1. 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.
2. 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....)

Cons:
1. 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
2. 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.
3. How about comparison with non-PINN methods, like classic simulation since the block-wise can be viewed as a kind of discretization.
4. the training time is still long and the cost for GPUs are large

I 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.

QUESTIONS:
1. the error seem to be not small (about 0.1), is that a so turblent case
2. the dimensionaless is great, so are there any plan to apply it for the real scale problem
3. how deos the inlet/outlet condition influence the problem or convergence?

LIMITATIONS:
N/A
TITLE: Official Review
CLARITY: 2
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-03

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe 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.\n\nSTRENGTHS AND WEAKNESSES:\nThe 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.\n\nQUESTIONS:\n- *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.\n\nLIMITATIONS:\nYes. 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.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-11"}