Title: UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss
Venue: NeurIPS 2025
Accepted: yes
Keywords: Mesh Adaptation​

Unsupervised Learning​

Partial Differential Equations

Computational Fluid Dynamic

Zero-shot Generalization
Abstract: Partial differential equations (PDEs) form the mathematical foundation for modeling physical systems in science and engineering, where numerical solutions demand rigorous accuracy-efficiency tradeoffs. Mesh movement techniques address this challenge by dynamically relocating mesh nodes to rapidly-varying  regions, enhancing both simulation accuracy and computational efficiency. However,  traditional approaches suffer from high computational complexity and geometric inflexibility, limiting their applicability,  and existing supervised learning-based approaches face challenges in zero-shot generalization across diverse PDEs and mesh topologies.
In this paper, we present an   $\textbf{U}$nsupervised and $\textbf{G}$eneralizable $\textbf{M}$esh $\textbf{M}$ovement $\textbf{N}$etwork (UGM2N). We first introduce unsupervised mesh adaptation through localized geometric feature learning, eliminating the dependency on pre-adapted meshes. We then develop a physics-constrained loss function, M-Uniform loss, that enforces mesh equidistribution at the nodal level. Experimental results demonstrate that the proposed network exhibits equation-agnostic generalization and geometric independence in efficient mesh adaptation. It demonstrates consistent superiority over existing methods, including robust performance across diverse PDEs and  mesh geometries, scalability to multi-scale  resolutions and guaranteed error reduction without mesh tangling.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presented an unsupervised mesh movement network that performs excellent generalizability across various PDEs and designed a M-uniform loss to constrain the equidistribution within mesh nodes. The method, named UGM2N, leverages a localized, patch-based approach inspired by vision transformers, which allows it to adapt meshes without requiring pre-adapted examples for supervision.\n\nSTRENGTHS AND WEAKNESSES:\nStrenth:\n\n1. **Strong experimental results showing excellent generalizability across various PDEs**.\n2. **Neat architectral design**. The decision to process local node patches instead of the entire mesh is an elegant and effective design choice.\n\nWeakness:\n\n1. **Clarity of key formulations.** The mathematical exposition could be improved for clarity. For instance, in Section 3.1, Equation (1) introduces a function $\\rho(x)$ within the integral definition of $σ =∫_K ρ(x) dx$. However, the function $\\rho(x)$ is never defined or explained in the text.\n2. **The theoretical justification for the M-Uniform Loss is not fully established.** In Appendix A.2, the proof that the proposed loss $L_M(θ)$ acts as a lower bound for the global mesh variance $L_{var}(M)$ hinges on the assumption that $Var(L_K) ≈ Var(L'_K)$ (line 465). This is justified by stating that each mesh element appears in a similar number of local patches, a condition that holds for regular meshes but is less certain for the highly irregular or complex geometries in real-world scenarios.\n3. **Lack of validation in large-scale, real-world scenarios.** The experiments (e.g., NACA airfoil, cylinder flow), are confined to 2D problems with relatively small mesh sizes (up to ~10k elements). Its performance on complex, industrial-scale 3D simulations, which often involve millions of elements remains untested.\n\nQUESTIONS:\n**model's true geometric invariance**. In Table 1, the Helmholtz equation test cases for the solutions cos(2πy) and cos(2πx) yield different Error Reduction (ER) values of 14.11% and 13.15%, respectively. Given that these two solution fields are mathematically symmetric. Could the authors elaborate on the potential cause of this discrepancy? Is this slight performance difference a result of inherent asymmetries in the training dataset, the underlying mesh triangulation, or another artifact of the learning process?\n\nLIMITATIONS:\n1. lack of real-world scenarios benchmarks.\n2. the writing of this paper is hard to follow.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present an unsupervised approach to adaptive mesh relocation using a residual graph transformer based mesh deformer. Trained with a patching strategy inspired by vision transformers and using a loss that equidistributes a monitor function inspired by Monge-Ampere. An iterative meshing strategy is used to push the mesh towards a global equidistribution equilibrium. Evaluation is performed on a range of PDE problems Poisson, Helmholtz, Burgers.\n\nSTRENGTHS AND WEAKNESSES:\nNovelty: I am concerned the main novelty of this work, the equidistributional loss function, coincides with a work that may be classifiied as a contemporaneous under the reviewer guidlines (see point 6. of https://neurips.cc/Conferences/2020/PaperInformation/ReviewerGuidelines). The authors cite Rowbottom et al \"G-Adaptivity: optimised graph-based mesh relocation for finite element methods\". They note that this work directly optimises against FEM error but miss that their $L_{equi}(Z)$ in equation (9) proposes the same equidistribution regularizer which is the main focus of this work in equation (7). Rowbottom et. al also use a patching based strategy inspired by vision transformers for scaling their approach. It should be noted however there are slight differences, Rowbottom et. al do not train solely on the equidistribution loss and in addition train on the full FEM truncation error. Moreover Rowbottom et. al use a global loss regulazier using a gradient based monitor function versus the current work that uses a local patch based Hessian monitor. A key detail is that the authors cite Rowbottom et. al July 2024 however a more recent version was uploaded to arxiv 6 Feb 2025 and is now public on ICML 2025 proceedings - https://icml.cc/virtual/2025/papers.html?search=G-Adaptivity:+optimised+graph-based+mesh+relocation+for+finite+element+methods. The work in July 2024 did not contain the equidistribution regularizer and based on the reviewer guidance, the arxiv-only work would be classified as \"non-refereed work\" and the ICML publication announcement falls inside the 2 months window.\n\nI would ask the AC to determine if the authors should be aware of this updated work, if so, then the submission lacks the sufficient novelety to be accepted at NeurIPS however if the determination is that this work need not be referenced then below I include my review of the paper on it's own merrit.\n\nThe proposed equidistributional loss is a powerful adaption, it allows GNN meshing approaches to be trained unsupervised without the require of expert meshes and aligns theoretically with classical Monge-Ampere methods. This can be seen by much stronger results compared to the supervised baselines M2N and UM2N and is further evidenced in the loss function ablation in Table 2. The novel patching strategy inspired by vision transformers allows for scaling to large meshes by paralysing over the domain.\n\nThe iterative mesh adaption may be a useful contribution allowing the mesh deformation to flow until convergence of equidistribution, resembling the parabolic relaxation of Monge Ampere methods. However there are some problems with it's presentation, it is clear how this is applied in inference, but in training it seems there is a distribution shift such that the mesh initialisation for later epochs approaches the target mesh deformation. The potential risk of training distribution shift is not discussed.\n\nIn evaluation Meshing time is not reported, which is common for baselines M2N/UM2N. This may be because the interative procedure involving interpolation is costly.\n\nIn the abstract the authors claim \"guaranteed error reduction without mesh tangling\" but I could not find any further discussion or expansion of this claim.\n\nQUESTIONS:\nN/A\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 1, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this paper, the authors propose an unsupervised and generalizable mesh movement network named UGM2N. The method utilizes a patch-based approach with an M-Uniform loss, which encodes local equidistribution properties. The proposed method is validated on various PDEs with different geometries and resolutions. The experiment results show that the UGM2N demonstrates generalizability and robustness abilities.\n\nSTRENGTHS AND WEAKNESSES:\n- Strengths\n\nThe patch-based idea and unsupervised M-Uniform loss are intuitive, and the results verify that they are effective in training a learning-based mesh movement model in an unsupervised manner.\n\n- Weaknesses\n\nThere are some missing evaluations. The proposed method is mainly applied to simple regular shape geometries such as circles and squares. It is not clear how the proposed method performs when applied to arbitrary boundaries in real-world cases. There is no discussion about how the proposed method handles mesh tangling or what the skill boundary of the UGM2N handling mesh tangling. The large variance of error reduction of the experiments is another concern. Please see more detailed comments in the question section. Would be happy to improve the score if the under-evaluation concerns are addressed.\n\nQUESTIONS:\n- In the mesh reconstruction process, are there any discontinuities around the patch boundaries? How the proposed method handle these potential issues?\n- Though the equal-distributed theory should make the mesh movement tangling free if the constraints are strictly satisfied, however, it is difficult for a learning-based method to achieve. In practice, how does the proposed method handle the mesh tangling issue? It will help to better understand the boundary of the proposed method if the authors can provide some tangled cases as well (e.g., similar to the tangling test in UM2N section 5.3)\n- How does the proposed method perform apply to real-world cases with arbitrary geometry boundaries? (e.g., similar to tsunami simulation cases in UM2N)\n- For the Poisson cases, the error reduction rates have a large variance between different u_exact,  what do the authors think induces this large variance?\n- For the time-dependent cases, does the proposed method always start to move a mesh from its uniform version, or the moved mesh is fed into the model autoregressive to next timestep?\n- For the moving cylinder and supersonic flow cases, how do the benchmark methods (MA, M2N, UM2N) perform?\n- Regarding the iterative mesh adaptation, how is the efficiency of the multiple iterations (including the cost of model inference and Delaunay Triangulation-based interpolation )? And just wondering why does the error reduction show a non-monotonic trend i.e., not a gradual convergence trend?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces UGM2N, an unsupervised mesh movement network for adaptive mesh generation in PDE solvers. It removes the need for supervised data by introducing a localized node-patch-based learning framework and a novel M-Uniform loss that enforces local mesh equidistribution. The method shows strong generalization across PDEs, mesh geometries, and resolutions, outperforming prior supervised methods (M2N, UM2N) and traditional Monge-Ampère-based techniques in many cases.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\nNovel unsupervised formulation: The M-Uniform loss and node-patch approach are original and practically motivated.\n\nStrong empirical results: Demonstrates excellent generalization across a wide range of PDEs and mesh configurations.\n\nEfficiency: The method remains computationally efficient despite its iterative adaptation scheme.\n\nWell-structured experiments: Includes diverse cases, ablation studies, and rigorous baseline comparisons.\n\nWeaknesses\nLimited theoretical depth: While the M-Uniform loss is well motivated, the theoretical analysis remains relatively basic (mostly variance decomposition) and lacks deeper convergence or approximation guarantees.\n\nBoundary limitation: The method currently ignores boundary nodes, which limits applicability to many real-world problems where boundary behavior is critical.\n\nAssumption on mesh regularity: The approach assumes quasi-uniform meshes where node degrees are roughly constant—this assumption may fail for highly anisotropic or unstructured meshes, but this limitation is not sufficiently addressed in practice.\n\nArchitecture simplicity: The graph transformer model used is relatively basic; it is unclear how much the performance comes from the loss function versus the model itself.\n\nQUESTIONS:\nProvide stronger theoretical analysis on convergence or optimality.\n\nEvaluate performance on highly irregular or anisotropic meshes to assess robustness.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-06"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents UGM2N, an unsupervised neural network for mesh movement in numerical simulations. The method introduces a \"node patch\" representation to process local mesh neighborhoods independently. It proposes a physics-constrained loss function, the M-Uniform loss, which enforces mesh equidistribution without requiring pre-adapted reference meshes for supervision. The goal is to create a generalizable framework that works across different partial differential equations (PDEs) and mesh geometries, improving simulation accuracy and efficiency.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The introduction of \"node patches\" is a novel approach for mesh adaptation. It processes the mesh locally, which simplifies the learning problem and is well-suited for parallel computation.\n- The method is unsupervised, learning from initial meshes and flow fields directly. This removes the dependency on generating high-quality adapted meshes with traditional, costly methods, which is a major practical advantage.\n- The experiments show the model's performance on varying mesh geometries, including different shapes (airfoil, cylinder) and resolutions, which is a good demonstration of its potential geometric flexibility.\n\nWeaknesses:\n- The paper has some presentation issues. Many specialized terms from computational physics and deep learning (e.g., \"Monge-Ampère based method\", \"GAT\", \"Delaunay triangulation\") are used without sufficient explanation, which makes the paper difficult to follow for an audience not expert in this specific field. I feel this is paper is not good venue for Neurips, more fitted to a physics journal.\n- The performance is not uniformly good. In Table 1, the method is sometimes outperformed by traditional methods (MA) or other learning-based baselines (M2N), which raises questions about the robustness of the proposed approach.\n- The experimental analysis feels incomplete. For example, there is no ablation study on key hyperparameters like the scaling constant λ in the proposed M-Uniform loss function.\n\nQUESTIONS:\n1. The M-Uniform Loss has a scaling hyperparameter λ set to 100. How was this value chosen, and how sensitive is the model performance to this value? An ablation study would be necessary to understand the robustness here. \n2. The M-Uniform loss calculation uses the Hessian matrix of the flow field solution u. This seems to require that the PDE expression is known. How can the method generalize to systems where the governing PDE is not known explicitly? If it cannot, I seriously question the generalizability claimed.\n3. In Table 1, the performance is sometimes weaker than baselines. For instance, in the first Burgers' equation case, M2N achieves 44.82% ER while your method gets 32.22%. Can you give some intuition for why the model does not perform as well in these cases? A clear explanation of failure modes would make me reconsider my score.\n4. The paper evaluates on complex geometries like cylinder flow and airfoil, which is good. However, the training is done only on four simple analytical flow fields. Is it possible the model overfits to certain features of these simple training functions? Seeing generalization from more complex training data would strengthen the paper.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-13"}