UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss
Authors: Zhichao Wang
Xinhai Chen
Qinglin Wang
Xiang Gao
Qingyang Zhang
Menghan Jia
Xiang Zhang
Jie Liu
Venue: NeurIPS 2025 poster ()
Keywords: Mesh Adaptation​
Unsupervised Learning​
Partial Differential Equations
Computational Fluid Dynamic
Zero-shot Generalization
Primary area: machine_learning_for_sciences
TL;DR: This paper proposes an unsupervised mesh movement network with a novel M-Uniform loss, achieving generalization across diverse PDEs and mesh geometries without labeled data.

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.

UGM2N: An Unsupervised and Generalizable Mesh Movement Network via M-Uniform Loss
Zhichao Wang 1.2 Xinhai Chen!.2x
Qinglin Wang Xiang Gao
Qingyang Zhangi.2 Menghan Jia Xiang Zhang1.2 1.2 Jie Liu
Laboratory of Digitizing Software for Frontier Equipment, National University of Defense Technology, Changsha, China, 2 National Key Laboratory of Parallel and Distributed Computing, National University of Defense Technology, Changsha, China, wangzhi chao@nudt. edu. en, chenxinhar 160nudt edu. cn
Abstract
Partial differential equations (PDEs) form the mathematical foundation for model- ing physical systems in science and engineering, where numerical solutions demand rigorous -efuaccyefficcency tradeoffs. Mesh movement techniques address this challenge by dynamically relocating mesh nodes to rapidly-varying regions, enhanc- ing 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. this paper, we present an Unsupervised and Generalizable Mesh Movement Network (UGM2N). We first introduce unsupervised mesh adaptation through localized geometric feature learning. eliminating the dependency on pre-adapted meshes. We then develop thhysics-constraing 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. 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.
Introduction
Solving partial differential equations (PDEs) fundamental for modeling physical phenomena. spanning fluid dynamics, heat transfer, quantum mechanics, and financial markets [1] Modern PDE solving critically relies on meshes, which serve as the foundational discretization framework for numerical methods 22] 3 The accuracy and computational cost of PDE solutions are significantly affected by mesh resolution: high-resolution meshes resolve complex physics at high computational expense, whereas coarse meshes improve efficiency but risk missing critical features. As problem complexity grows, geometric details demand exponentially finer resolution, while multi-physics inter- actions require dynamic adaptation ppashing memory, parallel efficiency, and solver convergence to their limits 4. To alleviate this issue, mesh adaptation methods- -arttcoulllat mesh refinement method (h-adaptation method) and mesh movement method (r-adaptation method) -dnamically op- timize computational resources, systematically overcoming traditional bottlenecks through intelligent spatial discretization control [5
Corresponding Author
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Mesh refinement method dynamically adjusts resolution via local element subdivissoo/ccoareening altering node counts while retaining fixed positions. In contrast, mesh movement method preserves node counts but relocates them strategically to high-resolution regions, guided by error estimators or gradients g While mesh refinement method handles discontinuities via topological changes, mesh movement method suits smooth domains, avoiding remeshing overhead. However, the traditional Monge-Ampère (MA)-based methods suffer from high computational costs due to (1) repeated mesh-motion PDE solves (e.g., solving auxiliary equations) and (2) mesh-quality checks to prevent inversion. In extreme cases, adaptive operations can exceed the PDE-solving cost itself, making the enhancement of mesh adaptation efficiency enduring open problem.
To improve the efficiency of mesh movement method, pioneering works employ supervised learning. training models on meshes adapted via traditional MA-based methods. Song et al. [ propose a mesh adaptation framework trained via MSE loss between initial and adapted mesh nodes, and Zhang et al. [10 introduce a zero-shot adaptive model trained with combined loss of volume preservation and Chamfer distance between initial and adapted mesh nodes. However, such supervised methods exhibit limited generalization: M2N requires PDE- and geometry retraining, risking mesh tangling under extreme deformations, while UM2N's zero-shot performance may degrade for unseen domains PDEs.
In this paper. we propose UGM2N, an unsupervised and generalizable mesh movement network. Inspired by vision Transformers [I,, we introduce node patches, locally normalized nodes with first-order neighbors, as model inputs. Unlike M2N/UM2N's whole-mesh processing, our method parallelly and independently computes adapted positions for each patch, simplifying the learning objective and enabling scale-invariant mesh adaptation. Leveraging the node patch representation, we formulate an M-Uniform loss function that mathematically encodes local equidistribution properties, the core objective of mesh movement methods. Minimizing this patch-wise loss can produce approximately M-Uniform meshes while effectively matching MA-based adaptation objectives-ale achieved through an efficient unsupervised framework. By learning adaptation dynamics directly, our model achieves equation-agnostic generalization while maintaining mesh-geometric independence.
Our main contributions are summarized as follows:
We present unsupervised mesh movement network, eliminating the need for pre-adapted meshes by learning solely on initial meshes and flow fields. Our novel node-patch represen- tation processes localized neighborhoods rather than full meshes, enabling efficient training and inherent generalization.
We derive a theoretically grounded M-Uniform loss function that enforces local mesh equidistribution at the node-patch level, which aligns with MA-based optimization objectives through a fully data-driven approach with native gquation-agnostr generalization across arbitrary mesh geometry.
We present extensive numerical validation showing exceptional generalizability and robust- ness across various PDE types (both steady-state and time-dependent), accommodating different boundary conditions initial conditions, and mesh geometries with varying shapes or resolutions.
2 Related Work
Machine learning for mesh generation and optimization. The automation and intelligence of mesh generation are among the key challenges in CFD 2030 [12), driving significant research efforts toward intelligent mesh generation and optimization. Zhang et al. [3, proposed the McshingNet and MeshingNet3D models to generate high-quality tetrahedral meshes, demonstrated in linear clasticity problems on complex 3D geometries. Chen et al. [15] introduced the MGNet model, which employs physics-informed neural networks [T6] to achieve structured mesh generation. For mesh optimization, Guo et al. [17], Wang et al. 188 19 developed intelligent mesh optimization agents based on supervised learning, unsupervised learning, and reinforcement learning (RL), achieving balance between optimization efficiency and quality.
Machine learning for mesh adaptation. Unlike static mesh optimization methods, mesh adaptation dynamically modifies the computational mesh during simulation to enhance resolution in critical regions (e.g., shock waves, boundary layers, dortex-dominated flows). These techniques are

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guided by error estimation schemes or feature-based criteria, ensuring computational efficiency while preserving accuracy. Advanced implementations leverage machine learning to predict optimal adaptation strategies, enabling high-fidelity simulations for complex, evolving flows.
For mesh refinement method, Foucart al. (20] pioneered RI. for adaptive mesh refinement, for- mulating it as POMDP and training policy networks directly from simulations. Dzanic et al. [21] developed DynAMO, using multi-agent RL predict future solution states for anticipatory refine- ment. Kim et al. (22) introduced GMR-Net, leveraging graph CNNs to predict optimal mesh densities without costly error estimation. Beyond these foundational works, research in intelligent h h-adaptive mesh refinement remains highly active, with additional advancements documented in [23 (9).
For mesh movement method, Omella and Pardo [30] proposed neural
boundary 9
node optimization method, which is specifically designed for tensor product meshes. Song et al.
introduced M2N, a framework combining neural splines with Graph Attention Network (GAT) 31
enabling and-to-end mesh movement with 3-4 wrder-of-magnitudee speedups. Hu et al. (32 introduced neural mesh adapter trained via the MA equation physical loss dynamically adjust mesh nodes, and develops a moving mesh neural PDE solver that improves modeling accuracy for dynamic systems. Rowbottom et al. [3] proposed a graph neural diffusion method that directly minimizes finite element error to achieve efficient mesh adaptation, and proved that its model architecture can effectively avoid mesh entanglement. For specialized applications, methods such as Flow2Mesh and Para2Mesh have demonstrated the efficacy of learning- bbaedd adaptation in aerodynamic simulations B4 35). Recent work extended these advances with UM2N [IO], a universal graph-tram tramfforrnf architecture attempts to achieve zero-shot adaptation across diverse PDEs and geometries. Most of the aforementioned works rely on supervised learning. where models are trained align their outputs with pre-adapted meshes, resulting in lack of physical information. Additionally, they often require retraining for different PDEs or mesh geometries, limiting their generalizability. This paper adopts an unsupervised learning approach to achieve equation-agnostic generalization across arbitrary mesh geometries.
3 Method
3.1
Problem statement and preliminaries
Given initial mesh M (e.g.. uniform mesh) and associated flow field variables (such as velocity or pressure p), the mesh movement method optimizes the node positions to generate an adapted mesh satisfying predefined resolution criteria. The mesh movement process can be analyzed from different perspectives, such as coordinate mapping between uniform and adapted meshes, uniform mesh construction in metric space, and so on G). The MA-based method adopts the former approach, solving the MA equation with boundary conditions to obtain the coordinate mapping (for detailed introduction, refer to App-Ay In contrast, this study employs the latter perspective, enforcing uniform distribution in metric space without explicit coordinate transformations between computational and physical domains.
From the latter perspective, given a physical domain Ω C Rdd (where d 2 1), the goal of mesh movement to construct uniform meshes in some metric space, which defined by a matrix-valued monitor function M = A (x), where x E Q. A mesh is said to satisfy the mesh equidistribution condition if it is uniformly distributed in this metric space, which can be mathematically expressed as:
J m(x)dx=
AK EM, N.
where m(x) dx, m(x) (x) is the mesh density function, K is the element of M, and N₂ is the number of clements in the mesh M. This condition constrains the size of mesh alements-wher m(x) is large, the element volume should be small, and vice versa. Additionally, the M-Uniform mesh condition also requires that the mesh elements should be equilateral in the metric space. In this work, we primarily focus modifying the mesh density while disregarding the equilateral alignment of mesh elements.
Compared to MA-based coordinate transformations, this approach for constructing uniform meshes in the metric-space space offers a more discretization-frient framework, particularly well-suited for local loss function modeling (see Section 3.3
3

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3.2 Network overview
The proposed UGM2N is illustrated Fig. [ The model takes the initial mesh with solution as input, and node patches are constructed from all mesh nodes, with input features generated using flow field variables. The coordinates mesh nodes within each patch are normalized to [0, 1] x [o, 1 via 0-1 normalization, then encoded through node and edge encoders to obtain embeddings. These embeddings are processed by multiple deform blocks and a node decoder, producing adapted node coordinates for each patch, which are denormalized to restore the original mesh. The flow field features the updated mesh are obtained through Delaunay Triangulation-bur -based interpolation on the original mesh, and the adapted mesh serves the initial input for the next iteration, with the process repeating for maximum of E epochs.
Thiae tocale
De'oe libit
Delane harpeiaio
Toli
Figure 1: The proposed mesh movement network.
Node patches. Inspired by vision Transformers [1]. the proposed model processes individual mesh node patches, unlike M2N or UM2N, which take the entire mesh input. This patch-based approach significantly improves local feature representation while maintaining computational efficiency and scalability, mirroring the local optimization principles employed in mesh smoothing techniques.
In each adaptive epoch, the input consists an initial mesh with flow field solution, denoted as M (V, E,E}, where V = {xi,... XN represents node coordinates, & denotes connectivity, and U = {u,,- UN } contains flow variables on the nodes. A node patch P; {X;, & defined as the node itself, its first-order neighbors, and their connections (excluding inter-neighbor connections). Patch normalization scales nodes to a unit square, reducing learning difficulty and accommodating varying mesh sizes. It is worth noting that normalization does not introduce additional computational overhead, as it can be efficiently implemented using vectorized operations.
The flow field features are incorporated into the patch features using mesh density function derived from the Hessian matrix:
M(x)) = 11 H(u)))
(2)
max, H ((g)))
[H(u)))
m(x:) = 1 max,j
[H()))|l
where is constant, = ||, la. I is the identity matrix, and |[H(w:)]] is the Frobenius norm of the Hessian. The scalar mxx)) ) is concatenated with node coordinates (yielding a 3D input for 2D meshes) as the model's input.
Mesh coordinate computation based on Graph Transformer model. The adapted node coordinates are computed using lightweight model. Node and edge features (central-to-ned neiihio coordinate vectors) are first encoded via dedicated MLP encoders, followed by L deform blocks for graph feature extraction. We employ despite simplicity. The adapted patch coordinates are then computed through the node MLP decoder, and the centering mesh node within each patch (the pink node in Fig. [ is denormalized to the original mesh space through vectorized operations. For boundary nodes, we ignore them and do not perform adaptation, since the output coordinates unlikely to lie precisely the boundary.
connectedd Graph Transformer [36] in each block, proving effective

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Iterative mesh adaptation with dynamic termination. Inspired by iterative mesh smoothing, multi- epoch mesh adaptation is employed to progressively refine node distribution during inference the Hessian norm values are updated between epochs via Delaunay riianulatiion-baer interpolation from original to adapted nodes, providing initialization for subsequent adaptations. While this process could theoretically continue indefinitely, convergence is not guaranteed and mesh validity may degrade (see App. B.2 for discussions). A fixed epoch count would limit optimization capability; instead, we propose metric-based adaptive strategy that dynamically determines termination based on optimization progress. This approach will be detailed following the presentation our unsupervised loss function.
3.3 M-Uniform loss
Unlike existing methods, which adopted supervised loss functions to align predicted meshes with reference meshes, our approach addresses two key challenges in practical applications: (1) the frequent unavailability of high-quality reference meshes, especially for multi-physics or geometrically complex problems, and (2) the poor zero-shot generalization novel PDE types beyond the training distribution. These limitations motivate our development of an unsupervised adaptation method.
As introduced in the Section 3.1 enforcing the mesh equidistribution condition offers novel approach. Eq. requires that the integral of mxx) over any mesh element K constant. However, since m(x) is only known at mesh nodes, exact integration is infeasible. We thus relax the strict M- Uniform condition to an approximate M-uniform condition:
f
m mx K m
Th
WKEM, M,
(4)
K
Ne
mkK = (de((M)), Mx = K / M(x)dx,
(5)
where K| represents the volume of the mesh element K and is a constant. Here, Mr is approximated via nodal averages: for triangular element K with nodes K1, K2, K3, Mr
وز M(xk) ) (note that M(x') require interpolation to obtain), where x² is the adapted position of node output by the model. Together with Eq. 2
and 3
we can obtain mk = 1/4 Eÿ= m(xk,) )
Building upon these foundations, we can define a metric function for element K:
Lx mx K||
(6)
Let Kÿ be the mesh element in the patch of mesh node i. Rewriting Eq, 4 in terms of the local mesh node i, we require that C be as uniform as possible around mesh node 2. We measure the variation in C ky among different mesh elements using a variance-baa bbased loss function:
N Lvar (Pi) = N Σ
[xx [(cx
Exz)
where N; is the number of mesh elements the patch Pis and Exp = proposed M-Uniform loss function can be writen as:
1 N, EN Lxg- Then, the
LM (0) = XE CV (P.),
(8)
where f is the model parameters, and 100 is a scaling constant. This approach shares conceptual similarities with PINNs, where local constraints-resinn residual conditions in PINNs and the M- M-Uniform condition here-guide the learning process. By enforcing mesh equidistribution condition at the node level, the model can adapt mesh node positions without supervised data, ensuring generalization to arbitrary adaptive scenarios. Crucially, unlike existing adaptive methods (e.g., M2N or UM2N), training does not require the full mesh as input. Instead, it can be trained on individual mesh nodes. For example, during and achieve efficient training through graph batching") Moreover, this approach further reduces the
training, we can sample a subset of mesh nodes from the mesh
2 Multi- mesh adaptation is disabled during training simplify the training process. 'Notably, during training, we do not use the model outputs to update the mesh nodes in other words, there is dynamic mesh update.
5

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required amount of data, as the number of data samples is proportional to the number of mesh nodes rather than the number of meshes.
During iterative mesh adaptation, we compute the global uniformity metric L (M') over the entire adapted mesh M' after each epoch assess equidistribution compliance:
N. Σ (Lx, Lxx)) Ne 1=1
1
Svar (M')=
where Ne is the number of mesh elements in M', and - +. Enn CK. The model stops the iterative mesh adaptation when Lvar (M') no longer decreases. During inference, we set a fixed upper we set the maximum number of mesh adaptation
N.
limit for the number of epochs as 10. The full adaptation algorithm is detailed in App. and the theoretical analysis of the effectiveness of the loss function to optimize mesh distribution is provided in App. B.2
Experiment
4.1 Experiment setups
Different numerical simulations involve diverse flow-field and mesh geometries characteristics. An effective mesh movement method must account for variations in both the flow field and the underlying mesh geometry. Our experiments show that the proposed method achieves robust, optimal performance across both
applied to different flow fields (diverse PDEs with varying solutions) or entirely distinct mesh geometries.
Model training. Following UM2N's protocol, we trained UGM2N on a mesh with only four flow fields and evaluated its zero-shot generalization performance unseen flow fields or meshes. As depicted in Fig. 12 (App. [C.T). random perturbations were applied to mesh node positions to enhance data diversity, resulting
training set comprising 10,440 mesh nodes. The model was optimized using Nadam (37 with an initial learning rate of le-4, with all experiments conducted on an NVIDIA RTX TITAN GPU. See App. D for more training details.
Baselines and metrics. We performed a comparative analysis against the MA method 38 M2N, and UM2N, using UM2N's pre-trained weights obtained from its GitHub repository (39] Perfor- mance was evaluated using two key metrics: (I) error reduction (ER), which measures the relative improvement in PDE solution accuracy compared the initial coarse mesh (with the high-resolution solution serving as the reference), and (2) tangling ratio (TR), defined as the fraction of invalid elements in the adapted mesh. Additional case-specific metries will be presented in the corresponding experimental sections. Detailed mathematical definitions of all metrics are provided in App. E.I
4.2 Performances across different flow field solutions
To assess the model's generalization ability across diverse flow fields, we conducted experiments using Burgers' equation with varying initial conditions, well as Poisson and Helmholtz equations with different analytical solutions (refer to App. C.3 and [n4][..ffor for more detailed PDE configurations). All simulations were performed on a uniform triangular mesh spanning the domain [0, 1] [0, 1], comprising 1,478 elements. For validation, high-fidelity reference solution was computed on a significantly refined mesh with 23,250 elements, ensuring precise accuracy for benchmarking purposes. The test results are summarized in Table [ can be observed that the our model demonstrates significant advantages in the vast majority of cases. In the seven test cases for the Poisson equation, our model achieved optimal performance (highest ER or lowest TR) in five cases, particularly excelling
with complex functions. For example, for Ex, exp[ 2x)) (y- (g-vea>² ours achieved an ER 0.25
0,25
of 9% far surpassing the comparison methods. Additionally, ours achieved complete dominance in the Helmholtz equation, securing the four highest ER out of five test cases. Although slightly inferior to M2N in the Burgers equation, ours still significantly outperformed UM2N. Moreover, in all cases, our model did not produce any mesh tangling phenomena (for the mesh tangling test on non-convex meshes, refer to App. E.2 These results validate the robustness and generalization capability of our

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Table Model performance of different flow fields
Variables
ER( TR(S).
MA
M2N
UM2N 40
Ours
Moxet
St Fono 22a one
15.40
0.92
6.74
14.56
En exp 0.25
025
-8.64
-30,20
-5.59
9.00
Polsson
(in( sin(4ry)
9.79
-98.01
-2.19
12.46
1/ e/ep(((e 0.5)
0.5)))
-28.22
1.15
-2.98
1.70
sin(2xa + 2xp)
-34,03
-9.07
9.07
cos(arz) exp( ( ((22 0.5) + 0.5)))
-8944
-41.89
5.15
4,90
0.3
0.57
-25.23
1.62
-3.53
2.56
expl ((x 0.5) (y 0.5)))
Mecae
ooel (09(2vg)
15.60
-11.16
10.86
14.11
10:29
-37.24
6.80
13.15
Helmholtz
ooe (2xy) cosl (mra))
13.48
-24.33
15.03
ons/ cool con(44ra))
10.87
-351.63
-2.61
14,09
ooes coe(4iry) (os(2mar)
Burgers
[suu (-20 (2 - 0..))). --33))))))))
26.81
44.82
0.46
32.22
exp
(1x 0.5) (y 0.51") 100)
51.12
29.93
22.76
30.19
method in mesh adaptation across different PDEs, achieving state-of-the-an performance compared
to existing approaches.
The mesh adaptation results for the Helmholtz equation are shown in Fig. 2 (for results on the Pois- son and Burgers equation, refer to App. E.3). Our method demon- strates superior visual alignment with the target flow field while simultaneously generating meshes with excellent quality compared to alternative approaches. is notewor- thy that, although M2N and UM2N can generate qualitatively adaptive meshes, they exhibit weaker com- pliance with the mesh equidistribu- tion condition compared with our method (see App. E.5 Addition- ally, in the present work the bound- nodes are kept fixed during adaptation; experiments that allow constrained movement of boundary nodes are reported in App. E.4
cor22ng)
XIA
UNIN
UGMGN
Figure 2: Mesh adaptation results for the Helmholtz equation with different solutions.
4.3 Performance across varying mesh geometries
Varying mesh resolutions. Practical simulations often involves meshes with varying element sizes, making it crucial for the model to handle meshes of different resolutions. We tested the model's performance on the Helmholtz equation with different mesh element sizes, where the solutions are the same as in Table [ The coarse mesh element sizes were [0.05, 0.04, 0.03, 0.02], corresponding to mesh element counts of [944, 1478, 2744, 5824]. As shown in Fig. 3 our model achieved improved solution accuracy across element sizes, whereas M2N model failed to adapt the mesh at any resolution, and UM2N could only generalize some of the element sizes. The results demonstrate
x, = [0.25, 0.25 y, [0.25, 0.25)

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C.44 201
Figure 4: The performance of mesh movement methods under different mesh resolutions is presented. The dashed line in the figure indicates the time required for UGM2N to complete a single iteration.
Figure 3: The ER for different mesh resolutions on the Helmholtz equation. For clarity, we clipped the minimum ER at -1,, even though for some methods (e.g., M2N), their adapted meshes significantly increased the solution error.
that the loss function based on MSE between mesh nodes in M2N struggles to generalize unseen meshes. Furthermore, UM2N's volume loss exhibits only limited generalizability.
Regarding computational efficiency, as illustrated in Fig. movement under varying element sizes. For each model configuration, we conducted ten repeated tests different solutions of the Helmholtz equation and reported the average mesh movement time per trial. Compared with the MA method, the neural-based mesh movement method demonstrates significant advantages. For detailed efficiency and scalability analysis, see App. E.6
present the time required for mesh
== 2.40s
= 2.53
3.5
2.50s
2.30s
30
Clarr nond
Figure 5: Results on the cylinder flow. We present the adaptive Figure 6: Mesh adaptation the meshes at four time slices, with the results over the entire simu- subsonic flow case. MA method lation period provided in the supplementary video materials. fails to converge in this case.
Varying mesh shapes. Another potential variable in the simulation is the shape of mesh. To evaluate the model's generalization capability under different mesh geometric configurations, quantitatively conducted three distinct simulation cases: subsonic flow around NACA0012 airfoil, cylinder flow, and the wave equation on a circular domain, with the experimental setups provided in App. C.3 The adaptive results of the airfoil mesh are shown in Fig. [6] which shows that our model effectively captures the shock wave location without introducing any invalid elements, Additionally, at the position 0.25, our model obtains the minimum mean absolute error (MAE in pressure coefficient result. For the cylinder flow case, neither MA nor M2N could produce valid meshes. As
'The average absolute error between the solution on the high-resolution mesh and the solution on the adaptive (coarse) mesh at each position (time step).

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shown in Fig. compared to UM2N, our model further reduces the prediction enror of the drag coefficient and slightly decreasing the cumulative error during the solving process. The results of the wave equation are shown in Fig. 7 At any given time step, our proposed method consistently generates smooth, high-quality adaptive meshes. In contrast, other methods may produce distorted mesh elements and fail to reduce errors, although they can also generate adaptive meshes.
To further validate UGM2N's generalization to more complex mesh topologies, we demonstrated its capability on irregular and anisotropic meshes in App. E.7 where the regularity assumption (App- B.2 may not hold. Additionally, we benchmarked all methods on 1,000 random polygonal domains using Gaussian mixture flow fields as exact solutions in App. E.8 UGM2N achieves Positive ER Ratio of 0.807 with stable mean ER of 13.99% 21.05% decisively outperforming MA
(0. 10) and UM2N (0. (0.245), both of which frequently yield negative ER. These results demonstrate that UGM2N achieves substantial error reduction and robust adaptation even in challenging scenarios.
Medi acceal Bx oytixder
Tinz agy
ER
= 0.738
1.25e
20
MA
FAIL
-8.18%
20
-52.75%
Figure 8: Mesh adaptation on moving cylinder and supersonic flow cases, In supersonic flow, UGM2N's adaptive mesh dy- namically adjusts nodes to track shock waves; in the moving cylinder case, precisely captures both the cylinder bound- ary and wake flow. The UM2N results for M2N are given in Figure 7: Mesh adaptation results on App. E.9 The results are also presented in the supplementary the wave equation.
21.81%
video.
Additionally, three qualitative peterimentt-supprseeee flow over a wedge, moving cylinder in a channel, and Tohoku tsunami with complex boundaries provided (first two in Fig. 8 third in supplementary materials), demonstrating effective adaptive meshes for realistic simulation scenarios.
4.4 Ablation study
Loss function To validate the effectiveness of the M-Uniform loss compared to the coordinate loss of M2N and the volume loss UM2N, we trained models with the same architecture and the same training data but different loss functions. The supervised data was generated using the MA method. presentss the average ER on PDEs with different solutions of models trained with different loss functions on three types of equations (the equation configurations With only a small amount of training data, supervised learning methods using the entire mesh as input struggle to produce effective models. In contrast, our unsupervised training approach requires no adaptive meshes as supervised data, and the node patch-based training method enables effective model training even with limited data. App. C.2 analysis shows increasing training data (10,440 to 41,760 patches) improves ER by up to 43% for Poisson and Helmholtz cases, indicating richer datasets further enhance model optimization.
the same in Table I
Iterative mesh adaptation demonstrate the effectiveness of our iterative mesh adaptation ap- proach, Fig. shows the error reduction across optimization iterations for the Poisson equation in

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[ The results reveal that the error reduction exhibits non mo-otonoc trend, initially increasing before decreasing as optimization progresses. Notably, our adaptive adaptation epochs (marked with a diamond) consistently stay within high ER regime, demonstrating the effectiveness of our method.
Neah Adaptation Spoch and ER
Table 2: The mesh adaptation performance of models trained
Folumn)
with different loss functions
Puluart
ER (%)
Loss
Poisson
Helmholtz
Burgers
Coordinate loss
-8. 19
-4.46
-9.17
Volume loss
-8.27
-0.52
-1.46
M-Uniform loss
5.21
9.94
30.07
Figure 9: Error reduction in solving the Poisson equation under different adaptive epoch settings.
Scaling parameter X. To assess the sensitivity of UGM2N to the scaling parameter in the M- Uniform loss, we conducted an ablation study with À = 10scale (scale E (-1. 1, 1,2,3), 5 runs per value). The left panel of Fig. 10 shows that the converged test loss scales linearly with A, consistent confirming that À modulates loss magnitude without affecting convergence dynamics. Adaptation performance, measured by error reduction, remains stable across most PDEs (right panel of Fig. 10p Minor sensitivity in Helmholtzl, Poisson4, and Poisson7 (highlighted in red) arises from case-specific optimization challenges, not model limitations, as these cases are difficult across all baselines. Overall, À exerts negligible influence on convergence and adaptation quality, validating the robustness of the M- M-Uniform loss design.
with Eq. 8
Test Lows vs
Adaptation Performance Across POES
Sondthee toase- -eeeeeeWW
17.5
Canvergee Teal Lowa
12-4
12.5
10-
7.5
2.5
1.0
Aureadd
Aureond
Caze
Figure 10: Ablation study scaling parameter X. Left: Converged test loss vs À (log-log scale) with theoretical linear fit (dashed). Right: Error reduction across PDEs; error bars represent ER variability across different \ values; red bars highlight sensitive cases.
5 Conclusion
We introduce UGM2N, an unsupervised and generalizable mesh movement network. This network removes the requirement for pre-adapted meshes while demonstrating strong generalization capabili- ties across various PDEs, geometric configurations, and mesh resolutions. By leveraging localized node-patch representations and a novel M-Uniform loss, our approach enforces mesh equidistribution properties comparable to Monge-Ampère- based methods but in more efficient, unsupervised man- ner. Extensive experiments demonstrate consistent performance improvements over both supervised learning baselines and traditional mesh adaptation techniques, achieving significant error reductions without mesh tangling across diverse PDEs and mesh geometries. See App. For the discussions on limitations and broader impacts.
10

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Acknowledgment
We appreciate the reviewers for their valuable insights and helpful comments. This research was partially supported by the National Key Research and Development Program of China (202IYFB0300001, 2023YFB3001903, the National Natural Science Foundation of China (12402349), the Natural Science Foundation of Hunan Province (2024J6668)) and the Youth Foun- dation of the National University of Defense Technology (ZK2023-11).
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A Monge-Ampère-basel mesh adaptation
Monge-Ampère (MA)-based mesh adaptation methods leverage optimal transport theory to map gradient Hessian information from the evolving physical solution onto mesh density function, thereby generating highly adaptive structured or unstructured meshes that dynamically refine in critical regions. The core idea is to reformulate mesh adaptation as solving the elliptic Monge-Ampor equation:
m(xx mg
dct dct(DD((x))
XEΩ
(10)
where mxx) is the monitor function, Mo is normalization constant. ((x) is the convex scalar potential whose gradient defines the optimal coordinate transformation, and o² is its Hessian matrix. This yields optimal transport mapping that automatically refines the mesh in high-gradient or high-curvature zones while coarsening in smooth regions.
However, solving the Monge-Amper equation demands costly nonlinear iterations- suchh as Newton or fixed-point methods-that are computationally expensive, sensitive to initial guesses, and often fail to converge in complex domains (e.g., non-convex geometries those with non-smooth boundaries). These limitations severely restrict the use of MA-based methods in real-time or large-scale adaptive simulations. The present work overcomes this bottleneck by introducing an unsupervised learning framework that accelerates the MA solution process while preserving high-quality adaptive meshes.
B
Method
B.1
The proposed mesh movement method
The proposed mesh movement method based on UGM2N presented Alg. [ Given an initial mesh and the corresponding flow field, UGM2N iteratively refines the mesh through multiple adaptive operations to obtain the optimal mesh. In this algorithm, operations such as Patch Processing, Mesh Reconstruction, and Convergence Check can all be vectorized, ensuring high computational efficiency in our method.
Algorithm The proposed mesh movement method
Input: Initial mesh M = (v*0, U U), max epochs
Output: Adapted mesh M
1: for to E do
2: Patch Processing:
3: Construct node patches (P; N from M
4: for each patch P. = XX, & do
5:
Normalize coordinates X; [0, 1]d
// Vectorized operations in the loop
6:
Compute density function mxx,) ) via and
7:
Encode features: H,
NodeEncoder X m (X,)))
// The operator is EdgeEEncoder((E)))
applied row-wise to the X;
8:
Update positions: X = DooeeDecoddr(tffmmBBlooo/////
9:
Denormalize centering node in X to the original mesh space
10:
end for
11:
Mesh Reconstruction:
Assemble adapted mesh = (V', U E
13:
Interpolate Hessian norm (or grad norm) |[H())|] via Delaunay triangulation
Convergence Check:
15:
Compute global uniformity Lvar (M') via Eq.
if Lvar stops decreasing or e E then
17:
M MM
18:
break
19:
else
M + M'
21:
end if
22:
end for
14

---

B.2 Analysis of M-Uniform loss
Here, we theoretically demonstrate that optimizing the local M-Uniform loss effectively optimizes the objective function associated with mesh equidistribution in one adaptation epoch. Consider a mesh MM (we omit epoch for clarity), let LK E Lyyy l {1,2, N.. E {1, ................................ No3 and L E {L,, E 21.2.... ..... Ne represent discrete random variables defined over mesh elements NN } be discrete uniform random variable. Here, N denotes the total number of mesh nodes, while N; indicates the number of mesh elements in the patch centered at node i. Simply put, k is random variable defined on all mesh elements, and L is a random
(see Eq. (nnd Eq. 9). and I E {1.2..... ...
variable defined on the mesh elements contained within all patches. According to the law of total variance, we have:
Var(Lx) = E[[aa[LK []]+ Var (B|LK) IJ).
(11)
The expectation E[Var Lx [)] quantifies the average local variance across node-centered patches, which simplifies to:
N
N N
N Laar (P:).
1 N
[Var (Lx [)]
Lkt - Txtt
(12)
Evae P2)
Moreover, from Eq.9 we have (M) Var(L'x). When the samples in Lı are repeated samples from L (i.e., each sample is duplicated times), we have Var(Lx) Var (xx) Assuming that each mesh element appears in approximately the same number of local patches i.e., the sampling of L'x in Lx is nearly identical-ww can adopt the approximation Var(Lx) Var(Lx). This regularity assumption holds, for example, in high-quality triangular meshes generated by mesh generation software, where almost all mesh nodes have degree of 6. Moreover, the number of nodes on the boundary is relatively small compared to the number of interior nodes. With Var (B[LK] 1]) 0, we get such an inequality:
N
1 N
1 J
L (M) VVr(L'x)) R Var(Lx) 2 E[Var(LK [1]]
Lvar (P:)
7 CM (0), (13)
i=1
XLrar 2 Las (0).
(14)
Therefore, (8) provides valid lower bound for (M)) In each adaptation epoch, when the model can successfully minimizes LMr(0) it concurrently optimizes the lower bound of XL thereby promoting mesh equidistribution. Moreover, in the limit where Lar(0) 0, all local variances Fvar (P:) vanish, implying that L,K is constant over all patches. Consequently, T'g must also be constant, leading to ACvar (M) O, which corresponds to exact mesh equidistribution.
Convergence and approximation analysis. The M-Uniform loss optimizes local mesh equidistribu- tion by minimizing Lvar (P,) (Eq. 7 similar traditional mesh smoothing techniques that optimize geometric quality metrics (e.g., aspect ratio). In a serial implementation, sequential node updates improve (Pa, for specific mesh node at each iteration, suggesting convergence. However, training errors limit exact minimization (Pt) = 0), and serial updates are computationally inefficient. Our Jacobi-style parallel updates enhance efficiency but complicate analysis due to data races, update conflicts, and the nonlinear nature of the transformation represented by our model. The non-convex Hessian of the objective function further precludes standard optimization guarantees. Despite these thcoretical challenges, we demonstrate through extensive experiments in Section 4 that UGM2N achieves robust convergence.
To further validate convergence, we track the nodal displacement norm 6, x k+1 X over iterations, where X denotes the mesh node positions at iteration k. As shown in Fig. 8x exhibits consistent decay toward zero across five Helmholtz equation cases, confirming convergence. Although &y may stagnate or even increase for larger iterations (k > 10) in some instances, our dynamic termination strategy (App. B.I effectively mitigates this, ensuring stable mesh adaptation.
15

---

Nodal Disple cement Nerm Across lberationn
2.12
X11
0.06
0.04
0.02
(k)
Figure 11: Nodal displacement norm 6, X h-+1
X across iterations for Helmholtz equations.
C
Dataset
C.1 Train data setups
We only used four flow fields on the same mesh to train the model, as shown in Fig. 12 The analytical solutions for the four flow fields are:
1. (z, y) 10 sin (mxx) sin(2
2. (z v) = å sin sin(TT) sin(Tyy;
3. (2 y) 10(sin(55z)) cos(10 25xy) cos(5x));
4. (x, y) 10(1 e cos(4Ty))
Additionally, we perform data augmentation on the mesh by introducing random perturbations to the mesh nodes.
1
1/2
04
Origin
Conneral
Figure 12: Flow fields for trianing the mesh movement model.
C.2 Impact of training data volume
UGM2N attains robust generalization using merely 4 flow fields (yielding 10,440 node-patch samples) via its unsupervised learning paradigm. To quantify the effect of training data volume, we evaluated 10,440, 20,880, and 41,760
model performance across 4, and 16 flow fields corressddng patches, respectively. As shown in Table performance improves steadily with increasing data volume. Notably, the Poisson and Helmboltz cases exhibit substantial error reductions, improving from 5.21% to 7.43% and 9.94% to 14.21% %, respectively. Overall, scaling the training data volume reliably boosts UGM2N's generalization capability, with especially marked gains complex PDEs, highlighting the critical role of larger datasets in future developments.
16

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Table 3: Error reduction (6) vs. training data volume (node patches)
Data Volume
Poisson
Helmholtz
Burgers
10,440
5.21
9.94
30.07
20,880
6.56
13.68
29.36
41,760
7.43
14.21
30.56
C,3 Test data setups
Helmholtz. The Helmholtz equation describes the propagation time-harmonic waves physics and engineering. Here, we solve equation of the following form:
-V + == f.u u on 0Q,
(15)
where u is the solution variable, an denotes the boundary, g is the boundary function for u, and is the source term. To test the generalization capability of the model, we construct five different solutions (see Table [ compute f and to formulate the Helmholtz equation, and evaluate the model's performance.
Poisson. The Poisson equation describes how scalar field responds to a given source distribution, written v f. Using the same approach as for the Helmholtz equation, constructed Poisson equations with seven different exact solutions (see Table T to test the model.
Burgers. The Burgers equation is fundamental nonlinear partial differential equation in fluid dynamics, combining convection and diffusion effects. this paper, we solve the following Burgers equation:
du at
+ (u V)u 2 0,
(16)
(n V u on Ω
where the viscosity coefficient 0.005, and the initial conditions are given in Table The simulation employs time step of At = 30 and runs for total duration of 0.5s.
simulated under conditions of Mach 0.8 at 1.55* angle of attack, while the cylinder flow case employed a Reynolds number of 100 with characteristic length of 0.2, kinematic viscosity of 0.01, time step of 0.001s, and total simulation duration of 3.5s. In our experiments, the airfoil case employed a coarse mesh with 10,466 elements and a high-resolution mesh with 41,364 elements, while the cylinder flow used 5,536 (coarse) and 11,624 (high-resolution) elements.
Airfoil and cylinder flow The airfoil
Wave equation. The wave equation is partial differential equation that describes the propagation of waves (such as sound waves, light waves, water waves, etc.) through a medium or space. This experiment solves the two-dimensional wave equation for the initial value problem unit circle, namely:
82u
v2u =0
(17)
012
where the initial condition is u1x y, 0) (1 - x² - -y²) sin(TT)) sin(Ty). The time step is 0.01s, and the total time 2s. The coarse mesh consists of 2,048 elements, while the high-resolution mesh contains 8,195 8,192 elements.
Moving cylinder and supersonic flow over the wedge. Both cases are benchmark cases from the Kratos official website [41]. The mesh element count for the moving cylinder case is 9,852, and the element count for the supersonic flow case 27,115. Please refer to the official site for more details.
C.4 Monitor function setups
Table 4 presents the values of the monitoring function a for different cases. Notably, in the airfoil case study, the monitoring function employs the gradient norm (rather than the Hessian norm) of mesh nodes to better capture the shock location. To address the long-tail distributions observed in both the cylinder flow and airfoil cases (since most mesh nodes have small Hessian norm values), logarithmic transformation was applied to the monitoring function at mesh nodes.
17

---

Table 4: a for different cases
Case
Train data
5
Poisson, Helmholtz, Burgers, Wave Cylinder flow, Airfoil case
5
10
D More model training details
We partitioned the dataset into training, validation, and test sets with an 8:1:1 ratio, employing the validation set for early stopping. The model architecture consists of node and edge encoders implemented as two linear layers [2, 512], followed by deformation blocks containing an 8-layer Graph Transformer with 512 hidden dimensions, residual connections, and attention heads, and finally a node decoder comprising
MLP [512, 256, 2]. All components utilize ReLU activation functions. The model was trained on machine with an 17-9700KF CPU. NVIDIA RTX TITAN GPU, and 64GB of memory, with the complete model only requiring approximately 3.1 hours of training time.
E
Results
E.1
Evaluation metrics
Error reduction (ER). Error reduction quantifies the relative enhancement in PDE solution accuracy. computed as the improvement over the initial coarse mesh solution, where the high-resolution result treated as the ground truth. For the steady case, error reduction is defined as:
Ugoarse
Uget
|12
Madapeed
Uret ll2 100%,
ER_
(18)
Ugcarse
Uref ll2
where Mcaure is the initial coarse mesh solution, uhr is the reference solution on the high- rigouresolution mesh, and Mndapted is the solution on the adapted mesh. A negative ER value indicates that the mesh adaptation process failed improve the solution accuracy compared to the initial coarse mesh. For the unsteady case, error reduction is used evaluate cumulative error reduction, defined as:
Uguarse,
Use
Ludaatedd
Uset, 1 x 100%.
ER
(19)
Uguarse,
where U i and Uref, represent the numerical solutions timestep i from the initial coarse mesh, adapted mesh and high-resolution mesh respectively.
Cumulative error. Cumulative error refers to the accumulation of emors over time, which is defined as:
Cumulative error
upet,ill:
(20)
where u, and Uref, the numerical solutions at timestep from the current mesh and the high- resolution mesh, and T is the total timestep.
MAE of Cp and Cp. The MAE (Mean Absolute Error) of the pressure coefficient Cp and drag coefficient Cp measures the errors between the solutions obtained different positions time steps and the results from the high-resolution mesh, defined as:
Cy,MAE Mean, Cpcf CD.MAF Mean;|Cn, Cief
(21)
where Cp. and CD represent the pressure coefficient and drag coefficient computed on the coarse (adapted) mesh at the i-th location (or time step), and Cre and Cref denote the corresponding values computed the high-resolution mesh at the same location/time step.
18

---

E.2 Mesh tangling evaluation on non-convex meshes
To further test whether our model avoids mesh tangling during the adaptation process on non convex meshes, we adopted the same testing method as UM2N [3). Specifically, we generated 100 meshes, each containing 8 nodes, with four located at the corners of a unit square and the remaining four randomly sampled within the unit square. The flow field was constructed using mixture of Gaussian distributions. The results are shown in Table [ demonstrating that both our model and UM2N consistently produced valid meshes. In contrast, the MA method struggles with non-convex cases and only generates valid meshes in limited scenarios. Fig. 13 shows some meshes after adaptation by our UGM2N model; our model generalizes well to unseen non-convex meshes without producing any mesh tangling.
Table 5: Results of the mesh tangling test on non-convex meshes
Metric
MA
UM2N
UGM2N
Tangling ratio per mesh (sta Valid mesh
10.919
0%
0%
41
100
100
MA
LMM2N
UGM2N
Figure 13: Mesh adaptation results of non-convex meshes (10 out of 100 cases).
tett (-aa)-("ar)""aaaa ("arr""(aaa)""
Vorr 129)
Poiston
Time
At
2At
3At
4AY
5At
GAA
7A
Bugara
(-20
50
0.57)
" foof (ir oss iv --aaa))) x
Figure 14: Mesh adaptation on Poisson's equation and Burgers' equation.
19

---

E.3
Mesh adaptation results on Poisson's 's equation and Burgers' equation
As shown in Fig. 14 our method can effectively generate adaptive meshes in flow fields with different distributions, whether for steady or unsteady cases.
E.4 Impact of boundary node treatment on model performance
Ensuring geometric consistency of mesh boundaries remains a key challenge in learned mesh movement. The core difficulty arises from inevitable prediction errors in ML models: minor deviations in predicted boundary node positions can severely distort boundary layers- thin near-wall regions where viscous, thermal, electromagnetic gradients are physically dominant and numerically critical.
To further evaluate different boundary handling strategies, we conducted experiments enabling simple clip-to-boundary projection (i.e., projecting adapted boundary nodes back onto the domain boundary if they drift outside), with results summarized in Table [ The results show that clip projection yields modest gains in internally dominated fows-where gradients or Hessian values are predominantly large in the interior-but delivers substantial improvements when large Hessian values are concentrated near the boundaries while interior values remain small (c.g.. Poisson4, Poisson6, Poisson7).
These findings confirm that, while our fixed- brundaby strategy ensures stability and prevents boundary layer collapse, simple post-hoc projection can safely enhance performance in scenarios where flow features vary sharply near boundaries. This highlights considerable potential for future improvements through learned boundary constraints or hybrid adaptation-projectin schemes, especially in wall- bounded and multi multi-physics applications.
Table 6: Error reduction with and without boundary clip projection. Cases follow Table in the main paper.
Burgers0
Burgers1
Helmholtzi
Helmholtz2
Helmholtz3
Helmholtz4
Helmholtz5
ER w.o. Clip (%)
32.22
30.19
14.11
13.15
15.03
14.09
16.98
with Clip (%)
35.27
30.18
16.68
16.17
15.94
15.57
17.34
Gain
13.05
10.01
12.57
13.02
10.91
11.48
10.36
Poisson1
Poisson2
Poisson3
Poisson4
Poisson5
Poisson6
Poisson7
ER Clip (%)
14.56
9.00
12.46
1.70
9.07
4.90
2.56
ER with Clip (%)
17.49
14.82
13.08
15.98
11.18
17.85
15.69
Gain (%)
12.93
15.82
10.62
114.28
12.11
122.55
113.13
E.5
The mesh equidistribution condition the adapted mesh
15 shows the absolute error between Lx and Zx on each mesh element after adaptation, while Table presents the value of Lvar (M') on the mesh. Quantitative analysis reveals our method produces minimal error values, indicating its adapted mesh most closely approximates the ideal cquidistribution condition.
Fig.
Table 7: Lvar (M') on the adapted meshes for the Helmholtz equation with different solutions
Solution of the Helmholtz equation
Method
cos(2Ty)
cos(2
cos(2mt) c2s()
cos(2ny) cos (aT()))))
cos(4ry) cos(2
MA 38
3.21e-7
3.89e77
1.68e-7
1.30e-7
1.50e-7
M2N
9
4.4le-7
5.90e-7
2.62e-7
4.67e-7
4.82e-7
UM2N
40
5.41e-7
5.50c-7
2.89c-7
3.32c-7
3.73e-7
UGM2N (ours)
2.12e-7
2.68e-7
1.52e-7
1.10e-7
1.19e-7
20

---

MA
MEN
UMZN
Figure 15: 1Cx Lxl on the adapted meshes.
E.6 Detailed efficiency analysis and scalability
key consideration is the scalability of our model to large meshes. Theoretically, our GNN-based model imposes no restrictions mesh scale, as it can process graphs of arbitrary size. However, practical GPU memory constraints limit single-GPU performance. On an RTX TITAN GPU (24GB), UGM2N can process up to 40,000 mesh elements, with inference requiring 345ms and interpolation 6s,s achieving a significant speedup over the MA method (25,510ms).
To further evaluate scalability, we measured inference, interpolation, and I/O times (data transfer between CPU and GPU) across varying mesh sizes (Table & can be observed that I/O operations account for the majority of the time overhead. This is because, in each epoch, the mesh must be transferred from the GPU back the CPU for Delaunay triangulation-bart accelerated Delaunay triangulation algorithm were available, both inference and interpolation could be performed entirely on the GPU represents one of the key directions for future optimization.
interpolation. a GPU-
Furthermore, UGM2N's absence of data dependencies between node patches enables efficient parallelization across N GPUs, with processing time scaling as O11/N) (excluding communication overheads). This supports real-time mesh adaptation for large-scale simulations.
Table 8: Timing breakdown (ms) for UGM2N across mesh sizes, For each element count, we repeated the experiment ten times and report the mean and standard deviation.
Elements
Inference
Interpolation
I/O
1,478
14.3
0.75+00.22
48.7=20.44
2,396
17.910.12
1.30
75.9
4,126
25.5
2.49-10.02
1364:0.39
9,248
43.911067
7.17400.7
302-10.83
36,098
354+1.88
60.0+
1,010
E.7
The mesh adaptation results on the irregular and anisotropic mesh
In App. B.2 we derive the validity of the proposed loss function by assuming mesh regularity
Var [xx Var(T'))) This assumption holds for meshes generated by mature mesh generation
software, they often ensure certain level of mesh quality (c.g., for 2D test case with 60k mesh
21

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elements generated by mature mesh generation software, we observed that 92.96% of nodes had degree of 6, while 2.96% had degree of 5 meaning around 95% of nodes fell into these two categories). Furthermore, we tested the model's applicability to irregular and anisotropic meshes-ia. -ie. scenarios where the assumption does not hold.
Irregular mesh. We uniformly sampled mesh points within a unit rectangular domain and generated meshes using Delaunay triangulation, followed by mesh smoothing to optimize quality, As shown in Fig. T6 the resulting meshes feature numerous nodes with varying degrees, thus failing to satisfy the regularity assumption. We generated a total of 10 such samples, with each sample's 's flow field solution defined as c2s(2 pz)), where Pu is random value uniformly drawn from [0, 1]. The error reduction results for our model are presented in Table 9 These results demonstrate that, despite being trained on regular meshes, our model generalizes effectively to highly irregular meshes, achieving substantial error reduction in 8 out of 10 cases.
Table 9: Error reduction of UGM2N on irregular meshes
Case
Case
Case 3
Case 4
Case 5
Casc 6
Case
Casc 8
Case 9
Case 10
ER (%)
-2.07
16.07
10.69
11.11
18.10
14.24
-2.10
8.46
9.84
7.16
Case
Cise
Origin
Adagted
Case
Case 8
Cise 9
Cisse
Adagted
Figure 16: Mesh adaptation results on irregular meshes (successful cases). can observed that mesh nodes move toward regions of high Hessian values in the flow field.
Anisotropic mesh. To evaluate our model's performance anisotropic meshes, we solve the Poisson (a 0.6)
equation with the exact solution u(x, y) (1- e
0.01
((x 0.5)² - for [z,, vl E [o, 1] x [o, 1].
We first generate preliminary stretched mesh elements via anisotropic adaptation produce the initial anisotropic mesh, as shown in Fig. 17 This mesh is then input to the mesh adaptation models, which further optimize the node distribution to align with the solution's characteristies. In this solution, Hessian values peak at approximately = 0.3. 0.5, and 0.7. As illustrated in Fig. 17 compared to the initial anisotropic mesh both the UM2N and UGM2N models more accurately capture these three locations, while the MA method yields invalid mesh. The absolute error reductions achieved after
22

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solving with the optimized anisotropic meshes are presented in Table 10 These results demonstrate that our model achieves the optimal error reduction, confirming its applicability to anisotropic meshes.
Uingia
LIMIN
UKIMIN
Figure 17: Mesh adaptation results Anisotropic meshes. The UGM2N model successfully captures the peak locations of the Hessian values.
Table 10: Absolute error reduction anisotropic mesh
MA
M2N
UM2N
UGM2N
Absolute error reduction
Fail
Fail
0.02
0.09
E,8 Large-scale robustness validation on random geometries
To rigorously evaluate generalization under extreme geometric and flow variability, we conducted large-scale experiment on 1,000 randomly generated mesh samples. Each sample features a random polygon with 3-6 edges and synthetic flow field drawn from a Gaussian mixture model. We solved the Helmholtz equation using these flow fields as exact solutions and compared mesh adaptation performance across methods.
Results are presented in Table 11 and Fig. 18 Both MA and UM2N exhibit high variance and frequent negative error reduction, reflecting poor generalization across diverse geometries and flow structures. In contrast, UGM2N achieves Positive ER Ratio of 0.807 (807 successful cases out of 1,000) with stable mean ER of 13.99% + 21.05%. demonstrating superior robustness. M2N was excluded from comparison due to its inability to generalize to arbitrary geometries and flow fields without case-specifie training.
Table 11: Robustness comparison 1,000 random polygonal domains with Gaussian mixture flow fields (Helmholtz equation). Positive ER Ratio = fraction of cases with ER > O.
Method
ER (mean % + x))
Positive ER Ratio
MA
-227.76 + 468.82
0.110
UM2N
-50.72 + 68.36
0.245
UGM2N (Ours)
13.99 + 21.05
0.807
E.9 More results of qualitative experiments
For the supersonic flow and moving cylinder cases, MA method exhibited divergence during the solution of the monitor function equation in both cases, ultimately failing to produce valid adapted meshes. As shown in Fig. 19] the M2N method displayed inconsistent behavior, particularly in the moving cylinder case, where it generated well-adapted meshes at certain time steps but yielded highly distorted elements others. For the supersonic flow, M2N successfully captured the shock position; however, the resulting mesh exhibited lower nodal density at the shock compared to UGM2N. Similarly, UM2N identified the shock location in the supersonic case but introduced undesirable element distortions in irrelevant regions away from the shock. In the moving cylinder scenario, while UM2N effectively fitted the flow field distribution, its excessive adjustments led prominent mesh distortions.
23

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Figure 18: Mesh adaptation results of UGM2N on different flow fields across different meshes (30 cases out of 1000). Whether changes the mesh or changes in the flow field, UGM2N demonstrates excellent robustness.
Limitations and broader impacts
Limitations. 1) The model currently fixes boundary nodes during adaptation to ensure geometric consistency and prevent boundary layer distortions; while post-hoc clip projection to the boundary yields modest improvements some cases, future work could integrate learned boundary constraints for more flexible handling. 2) The approximation in Eq. 13 assumes mesh regularity (i.e., similar node degrees), but empirical tests on irregular meshes (App. E.7) demonstrate robust performance in 80% of cases despite violations. Further validation on highly non-uniform or anisotropic meshes with extreme degree variations remains warranted. 3) The model adopts a relatively simple and lightweight architecture. Future work could explore more complex model designs to achieve better mesh adaptation performance. 4) By adopting a node-patch-basse processing strategy, our method achieves scale invariance and translation eivariance-that is, adaptation results remain unchanged under uniform scaling or translation of the mesh. However, we observe that Al-based mesh adaptation methods currently lack rotation invariance: rotating the coordinate system of the mesh leads to inconsistent results. This remains an important open challenge for future work.
Broader impacts. The UGM2N method proposed in this paper significantly improves the efficiency of mesh movement techniques in mesh adaptation through unsupervised learning and localized mesh adaptation technology. It reduces the high computational costs of traditional mesh adaptation methods, thereby accelerating the simulation of complex physical phenomena such fluid dynamics and heat transfer under limited computational resources. Its generalizability allows it to be applied to various partial differential equations and geometric shapes, enhancing the practical engineering applications of current Al-based mesh adaptation methods.

---

Time stop
UMEN
10
40
60
Time step
UMZN
Figure 19: Mesh adaptation results of the M2N and UM2N models for the supersonic flow and moving cylinder cases. Red boxes mark the locations of mesh distortions.
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Reviews:
[{"review_id": "uaRIFnEnuE", "reviewer": "Reviewer_mhZx", "summary": "This 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.", "questions": "1. 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.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "The reviewers have addressed my concerns in their rebuttal, and they have promised to improve the writing for the final version.", "rebuttal": "We sincerely appreciate the reviewers' insightful comments and constructive suggestions, which have greatly helped us improve the quality of our paper. Below, we will address each of the weaknesses (**W**) and questions (**Q**) raised in your review comments point by point.\n\n### **Response to W1**\n\n**Terminology**\n\nWe fully acknowledge your concern regarding insufficient explanations of specialized terms. Since our work involves AI for physics, the field encompasses a wide range of specialized terminology due to its interdisciplinary nature, combining concepts from physics, applied mathematics, and computer science. We recognize that additional context would benefit readers with diverse backgrounds. In the revised version, we commit to expand explanations in the main text or appendices  and include targeted references  for further reading.\n\n**Suitability**\n\n *Machine learning for physics-based simulation* is a thriving subfield within *AI for Science*, as evidenced by numerous seminal works published at NeurIPS (e.g., *SchNet*, *Hamiltonian Neural Networks*, and *Neural ODEs*). The prior AI-based r-adaptation works, *M2N*  and *UM2N*, also appeared in NeurIPS. We believe this work aligns well with NeurIPS’s scope.\n\n### **Response to W2**\n\n**Robustness**\n\nWe acknowledge that our model does not achieve optimal performance in every single case. However, as evaluated from multiple perspectives in the paper, our approach consistently outperforms existing methods across various scenarios, including:\n\n1. **Flow Field Variations:**\n    - Achieves **9 best results out of 14 cases** (64.3%).\n    - Outperforms existing ML-based methods (*M2N* and *UM2N*) in **12 out of 14 cases** (85.7%).\n2. **Mesh Scale Adaptability (Fig. 3)**:\n    - Maintains error reduction capability across different flow fields.\n    - Delivers **14 best results out of 20 test cases** (70%).\n    - Surpasses ML-based baselines in **15 cases** (75%).\n3. **Variable Geometry Tests**: Exceeds both traditional MA methods and ML-based approaches.\n\nThese results confirm that **our method offers the most consistent performance overall**, despite minor limitations in isolated cases. We also want to emphasize that achieving perfect performance in *every* scenario is inherently challenging—even traditional methods fail in certain cases (see **Tab. 1**).\n\n**Additional Robustness Validation**\nTo further validate robustness, we have conducted a new large-scale test on 1,000 mesh samples, where each sample contains random polygonal shapes (3–6 edges) and Gaussian mixture-based flow fields.\n\nWe solved the Helmholtz equation using these flow fields as solutions, with the comparative results shown in the table below. The results show that both MA and UM2N suffer from performance variance across different flow fields and geometries. In contrast, our method demonstrates significantly greater robustness in mesh adaptation.\n\nNotably, our approach achieves the highest Positive ER Ratio (807 successful cases out of 1000), substantially outperforming both he traditional method and UM2N. The M2N results have been excluded from the table due to its inherent limitation in generalizing across arbitrary geometries and flow fields without case-specific training.\n\n| Method           | Error Reduction (mean%±std%) | Positive ER Ratio |\n| - | - | - |\n| MA               | -227.76±468.82 | 0.110       |\n| UM2N             | -50.72±68.36 | 0.245      |\n| **UGM2N (Ours)** | **13.99±21.05** | **0.807** |\n\n### **Response to W3\\Q1**\n\nThank you for pointing out the limitations in our experimental design. We have conducted an ablation study to systematically evaluate the influence of parameter $\\lambda$ on model performance. We define $\\lambda$ = 10**scale, with scale values spanning the range from -1 to 3 in increments of 0.5. To ensure  reliability, each $\\lambda$ value was tested with 5 repeated experiments. The converged test losses are shown in the table below, which indicate a linear relationship between $\\lambda$ and the final converged loss, which aligns with the Eq. (8) of our loss function. This suggests that the value of $\\lambda$ has minimal impact on the model's convergence.\n\n| $\\lambda$      | 0.1                  | 0.32                 | 1                    | 3.2                  | 10                   | 32                   | 100                  | 320         | 1000        |\n| - | - | - | - | - | - | - | - | - | - |\n| **Test Loss** | (3.89 ± 1.52) × 10⁻⁴ | (1.21 ± 0.22) × 10⁻³ | (4.12 ± 0.98) × 10⁻³ | (1.07 ± 0.08) × 10⁻² | (4.33 ± 2.38) × 10⁻² | (1.73 ± 0.77) × 10⁻¹ | (4.11 ± 1.28) × 10⁻¹ | 1.35 ± 0.47 | 3.20 ± 0.24 |\n\nThe results on the  the Poisson and Helmholtz equations are summarized in the below table. Overall, the ER remain relatively stable across different $\\lambda$ values, suggesting that our loss function is not highly sensitive to $\\lambda$. However, certain cases (e.g., Helmholtz1, Poisson4 and Poisson7) exhibit greater sensitivity to $\\lambda$, which we attribute to case-specific effects rather than model limitations, as they prove challenging to optimize across all models.\n\n| PDE    | Helmholtz1   | Helmholtz2   | Helmholtz3   | Helmholtz4   | Helmholtz5   | Poisson1     | Poisson2    | Poisson3     | Poisson4    | Poisson5    | Poisson6    | Poisson7    |\n| - | - | - | - | - | - | - | - | - | - | - | - | - |\n| ER (%) | 13.34 ± 4.48 | 10.89 ± 1.74 | 14.74 ± 0.72 | 13.23 ± 0.63 | 15.53 ± 1.03 | 13.06 ± 2.03 | 8.82 ± 0.68 | 12.17 ± 0.70 | 2.28 ± 1.26 | 9.15 ± 1.36 | 5.40 ± 1.65 | 2.87 ± 1.26 |\n\n### **Response to Q2**\n\nThis is indeed an important point that warrants clarification. In solving PDEs, we typically know the governing  PDE and boundary/initial conditions, while the solution *u* remains unknown. As you correctly noted, for simple problems we may obtain explicit expressions for *u*, but for complex flow problems such explicit solutions are generally unavailable.\n\nHowever, our method does not require explicit expressions for *u* to compute the Hessian matrix, since we can numerically recover solution derivatives through established numerical differentiation techniques, which is detailed in book *Adaptive Moving Mesh Methods* P266. This numerical method only requires flow field variables at mesh nodes and computes the Hessian (and gradient) values via the least squares fitting method or the Galerkin formulation method, enabling our approach to generalize to arbitrary flow fields.\n\n### **Response to Q3**\n\nThis is an excellent question that provides valuable insights for further model optimization. Our analysis reveals that M2N's superior performance on the first Burgers equation stems from its tendency to generate more deformed meshes (compared to our smoother UGM2N outputs), which better capture abrupt flow variations in this specific case. However, as Fig. 2 demonstrates, this characteristic often leads to mesh distortion in most other cases, ultimately compromising accuracy. This suggests M2N's advantage is case-specific rather than methodological.\n\nRegarding the suboptimal results on Poisson4-7, we identified that our boundary treatment - specifically, not updating boundary nodes - becomes problematic when large Hessian values concentrate near boundaries while interior values remain small. New experimental validation (see table below) shows significant improvement when allowing boundary-constrained node movement. Notably, even without this modification, our model outperforms both M2N and UM2N (which natively permit boundary movement), confirming our approach's fundamental advantages.\n\n| PDE    | Poisson1 | Poisson2 | Poisson3 | Poisson4 | Poisson5 | Poisson6 | Poisson7 |\n| - | - | - | - | - | - | - | - |\n| ER (%) | 17.49    | 14.82    | 13.08    | 15.98    | 11.18    | 17.85    | 15.69    |\n\nFurther tests reveal boundary adaptation's differential impact:\n\n- **Helmholtz cases**: Minimal improvement as flow variations predominantly occur in interior regions\n- **Burgers cases**: Substantial ER boost to 35.27% in Case1 but limited gains in Case2\n\nThese findings highlight two critical factors: (1) solution characteristics and (2) boundary treatment quality, particularly for curved boundaries where simple assignment proves inadequate and projection becomes necessary. Developing enhanced boundary handling methods for complex geometries emerges as a key direction for future research.\n\n\n### **Response to Q4**\n\nWe appreciate your thoughtful comments. First, while our model was trained on *only 4 flow fields*, the use of node patches as fundamental data units resulted in a substantial dataset of 10,440 samples, which proved sufficient for effective model training. The model's demonstrated generalization capability across multiple test cases, enabled by its few-shot learning ability, eliminates the need to prepare large-scale training datasets.\n\nTo further investigate the impact of training data volume on model performance, we have conducted additional experiments with expanded flow field configurations (4, 8, and 16 fields), corresponding to training nodes of [10440, 20880, 41760]. The experimental results demonstrate consistent improvement in our model's generalization capability with increasing training data across Poisson, Helmholtz and Burgers equations, with particularly significant gains observed for Poisson and Helmholtz cases. This inspired us to expand richer datasets in subsequent work to further enhance the model's generalization ability.\n\n| Data number | Poisson | Helmholtz | Burgers |\n| - | - | - | - |\n| 10440       | 5.21    | 9.94      | 30.07   |\n| 20880       | 6.56    | 13.68     | 29.36   |\n| 41760       | **7.43** | **14.21** | **30.56** |\n\n------\n\nWe sincerely thank the reviewers for their thoughtful comments and insightful questions. Your feedback has been invaluable in helping us reflect on and refine various aspects of our work. We look forward to any further discussion or clarification you may require."}, {"review_id": "3E5iWLt7OA", "reviewer": "Reviewer_Htea", "summary": "This 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.", "questions": "Provide stronger theoretical analysis on convergence or optimality.\n\nEvaluate performance on highly irregular or anisotropic meshes to assess robustness.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "great rebuttal and thanks for the efforts on the improvement; i have raised my score.", "rebuttal": "The authors gratefully acknowledge the reviewers' valuable feedback, which helped us refine and strengthen the paper. Below we provide our responses to the reviewers' comments, where questions are labeled as **Q** and weaknesses are denoted as **W**.\n\n### **Response to W1\\Q1**\n\nWe sincerely appreciate the reviewer’s valuable suggestions regarding convergence and approximation guarantees. We fully acknowledge the concerns and recognize that Appendix A.2 currently provides only a preliminary theoretical analysis. As noted in our paper, our method draws inspiration from conventional mesh smoothing techniques, which also optimize mesh quality by minimizing a scalar function defined over local elements. However, our approach differs fundamentally in that we optimize the monitor function integral approximation $\\mathcal{L}\\_{K} =m_{K}|K|$  (Eq. 6), whereas traditional smoothing focuses on geometric quality metrics (e.g., aspect ratio, minimum angle).\n\nWhile this similarity might suggest that convergence or approximation guarantees could be readily derived, such analysis proves challenging in practice. Under a serial implementation—where nodes are updated sequentially—each iteration provably improves $\\mathcal{L}\\_{\\mathrm{var}} ( \\mathcal{P}\\_{i} )$ within a local neighborhood, theoretically ensuring convergence. However, the final approximation accuracy is inherently limited by the model’s training error, as the learned mapping cannot guarantee exact minimization (i.e., $\\mathcal{L}\\_{\\mathrm{var}} ( \\mathcal{P}\\_{i} )=0$) for arbitrary node patches. Moreover, this serial approach is computationally inefficient due to its non-vectorizable nature.\n\nIn contrast, our practical implementation employs Jacobi-style parallel updates, which significantly improve efficiency but introduce inherent analytical difficulties. Although the model effectively optimizes local node distributions, the convergence behavior and approximation error remain challenging to characterize theoretically. Unlike Laplacian smoothing, which permits linear analysis, our transformation $T$ operates as a black-box nonlinear function, making it challenging to verify whether it possesses certain properties (e.g., whether it constitutes a contraction mapping).\n\nFurther complicating the analysis are the practical challenges of parallel updates, including data races (concurrent modifications of shared regions), update conflicts (local improvements causing global degradation), and information delays (reliance on stale neighbor positions). Additionally, while local node neighborhoods are often geometrically convex, the Hessian of our objective function may not be, precluding standard convex optimization arguments.\n\nThese challenges reflect broader theoretical difficulties inherent to adaptive mesh optimization, and are not unique to our model. Nevertheless, as demonstrated empirically in Section 5, our method achieves robust convergence and superior adaptation quality in practice. We have expanded Appendix A.2 to further clarify these theoretical considerations in response to the reviewer’s insightful feedback.\n\nDespite the aforementioned theoretical challenges, we have empirically evaluated the convergence behavior of our model by monitoring the evolution of the nodal displacement norm  $\\delta_k=||\\mathbf{X}^{k+1}-\\mathbf{X}^{k}||$ across iterations, where $\\mathbf{X}^{k}$ denotes the mesh node positions at optimization step $k$. As demonstrated in the results for the Helmholtz equations (shown below), the consistent decay of $||\\mathbf{X}^{k+1}-\\mathbf{X}^{k}||$ toward zero with increasing $k$ provides empirical evidence of algorithmic convergence. However, when $k$ increases (greater than 10), we also observed the phenomenon that $\\delta_k$ fails to converge or even increases in test cases. In such scenarios, our proposed dynamic termination method can effectively address this issue, thereby enhancing the model's convergence.\n\n| PDE        | $\\delta_0$ | $\\delta_1$ | $\\delta_2$ | $\\delta_3$ | $\\delta_4$ | $\\delta_5$ | $\\delta_6$ | $\\delta_7$ | $\\delta_8$ | $\\delta_9$ |\n| - | - | - | - | - | - | - | - | - | - | - |\n| **Helmholtz1** | 0.121      | 0.089      | 0.070      | 0.060      | 0.053      | 0.047      | 0.045      | 0.042      | 0.045      | 0.047      |\n| **Helmholtz2** | 0.110      | 0.081      | 0.068      | 0.058      | 0.051      | 0.046      | 0.042      | 0.039      | 0.036      | 0.034      |\n| **Helmholtz3** | 0.105      | 0.075      | 0.058      | 0.047      | 0.038      | 0.032      | 0.028      | 0.026      | 0.024      | 0.022      |\n| **Helmholtz4** | 0.120      | 0.062      | 0.041      | 0.032      | 0.028      | 0.025      | 0.024      | 0.022      | 0.022      | 0.022      |\n| **Helmholtz5** | 0.122      | 0.060      | 0.038      | 0.029      | 0.025      | 0.023      | 0.023      | 0.023      | 0.023      | 0.023      |\n\n### **Response to W2**\n\nWe sincerely appreciate your valuable feedback. When dealing with complex boundaries, ensuring the geometric consistency of mesh boundaries is a challenge. The fundamental difficulty stems from the inevitable prediction errors in ML models, where even minor deviations of predicted boundary nodes can significantly compromise accuracy by displacing boundary layers - the critical thin regions near surfaces where viscous, thermal or electromagnetic effects dominate. This justifies our design choice to avoid direct boundary node prediction.\n\nOur new experimental validation reveals that while simple clip operations (projecting nodes onto boundaries) show minimal impact (list below) for internally-dominated flows like Helmholtz and Burgers cases in our paper, they provide substantial improvements for cases involving rapidly varying boundary conditions (detailed in Response Q3 to Reviewer mhZx). These findings indicate that, while our current treatment provides reasonable robustness, there remains ample opportunity to further improve boundary prediction through learned or hybrid approaches.\n\n| PDE        | Burgers0 | Burgers1 | Helmholtz1 | Helmholtz2 | Helmholtz3 | Helmholtz4 | Helmholtz5 |\n|-|-|-|-|-|-|-|-|\n| **ER w.o. Clip (%)** | 32.22 | 30.19 | 14.11 | 13.15 | 15.03 | 14.09 | 16.98 |\n| **ER with Clip (%)** | 35.27    | 30.18    | 16.68      | 16.17      | 15.94      | 15.57      | 17.34      |\n\n### **Response to W3\\Q2**\n\nThank you for your feedback. Following your suggestions, we have added two new experiments to evaluate the performance of our method on both highly irregular and anisotropic meshes. The detailed experimental setup and results are presented below.\n\n**Highly irregular mesh** \n\nWe uniformly sampled mesh points within a unit rectangular domain and generated meshes via Delaunay triangulation, followed by mesh smoothing to optimize mesh quality. The meshes generated this way contain numerous mesh points with different degrees, making them fail to satisfy mesh regularity.\n\nWe generated 10 samples in total, with the flow field solution on each sample given by $\\cos(2  \\pi  (x - \\mu_x))$, where $\\mu_x$ is a random value in [0, 1]. The error reduction results of our model are shown below:\n\n|            | Case 1 |   Case 2   | Case 3 |  Case 4   | Case 5 |  Case 6  |    Case 7 |    Case 8 |  Case 9   |   Case 10  |\n| ---------- | ------ | ---- | ------ | ---- | ------ | ---- | ---- | ---- | ---- | ---- |\n| **ER (%)** | -2.07 | 16.07 | 10.69 | 11.11 | 18.10 | 14.24 | -2.10 | 8.46 | 9.84 | 7.16 |\n\nThe results demonstrate that, although our model was trained on regular meshes, it can generalize well to highly irregular meshes, achieving effective error reduction in 8 out of 10 cases.\n\n**Anisotropic meshes**\n\nWe have added a new test case for anisotropic meshes, where the flow field solution is given by $u(x,y)=(1-e^{\\frac{-(x-0.5)^2}{0.01}})((x-0.5)^2-1), [x,y] \\in [0,1] \\times [0,1] $.  First, we generate preliminary stretched mesh elements through anisotropic adaptation to create anisotropic meshes. These generated anisotropic meshes are then fed into our model to further optimize the node distribution to match the characteristics of the solution. In this solution, the Hessian values reach their maxima at approximately x = 0.3, 0.5, and 0.7. Our UGM2N model can capture these three locations more accurately compared to the initial anisotropic meshes. After solving with the optimized anisotropic meshes, the error reduction (absolute value) for each model is shown in the table below, demonstrating that our model has the potential to be applied to anisotropic meshes.\n\n|                 | MA   | M2N  | UM2N | UGM2N    |\n| --------------- | ---- | ---- | ---- | -------- |\n| Error reduction | Fail |Fail      | 0.02 | **0.09** |\n\n### **Response to W4**\n\nWe sincerely appreciate your valuable feedback. In fact, our ablation studies in Section 4.4 have already evaluated the impact of different loss functions on model performance under identical architectures. The results demonstrate that our proposed loss function outperforms those used in previous methods when maintaining the same network structure. We would be happy to conduct further comparisons of loss functions and architectures (including M2N and UM2N) to validate our approach, if of interest.\n\nFurthermore, compared to prior approaches, our model architecture is significantly simpler. This design choice enables highly efficient inference per adaptation epoch (as shown in Figure 4), while maintaining substantially better computational efficiency than MA methods even for multiple adaptation rounds.\n\n------\n\nWe greatly appreciate the time and effort the reviewers have devoted to evaluating our work. Your constructive feedback has deepened our understanding of the key issues and has been extremely helpful. We remain open to further suggestions and look forward to your continued guidance."}, {"review_id": "NBMfMDbSLd", "reviewer": "Reviewer_AXEC", "summary": "In 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.", "questions": "- 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?", "limitations": "yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "- 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.", "was_revised": true, "final_justification": "The authors’ rebuttal addressed my concerns. I would like to raise my score to borderline accept.", "rebuttal": "Thank you for your  constructive comments on our paper. We sincerely appreciate the  effort you have dedicated to reviewing our work. Below, we address each of your questions and concerns in detail, with clarifications, new experiments, and implementation insights, where **Q** represents the question and **W** represents the weakness. We hope that our clarifications and additional experiments may encourage you to reconsider your score.\n\n### **Response to Q1**\n\nWe apologize for the unclear description of the reconstruction process. According to Fig. 1, our method appears to reposition all mesh nodes (including boundary nodes) of each patch back onto the mesh; however, as stated in *Line 165* and *Algorithm 1*, we actually only reposition the central nodes, thereby avoiding issues arising from patch boundary conflicts. \n\n### **Response to Q2**\n\nWe greatly appreciate your constructive feedback. First, we would like to address the issue of mesh tangling. Our model utilizes an iterative mesh adaptation process; however, we observed that excessive iteration counts lead to progressive instability and the generation of tangled elements. To resolve this issue, we have implemented a monitoring mechanism for $\\mathcal{L}_{\\mathrm{var}}\\left(\\mathcal{M}^{\\prime}\\right)$ on the mesh and adopted dynamic termination to prematurely halt the adaptation process. \n\nSecond,  we have conducted the tangle test you recommended in Section 5.3 of UM2N for further validation. We generated 100 meshes, each containing 8 nodes, with four located at the corners of a unit square and the remaining four randomly sampled within the unit square. The flow field was constructed using a mixture of Gaussian distributions. The results are presented below, demonstrating that both our model and UM2N consistently produced valid meshes. In contrast, the MA method struggles with non-convex cases and only generates valid meshes in limited scenarios.\n\n|                           | MA            | UM2N  | UGM2N |\n| - | - | - | - |\n| Tangling ratio per mesh (mean±std) | 6.63%±10.91% | **0%** | **0%** |\n| Valid mesh | 41 | **100** | **100** |\n\n### **Response to Q3**\n\nThis is an extremely constructive suggestion. We have added the Tohoku tsunami simulation case adopted in UM2N. The results show that:\n\n1. Our model accurately captures wave propagation, with mesh element sizes dynamically adapting to the wave movement;\n2. In this complex boundary simulation, our model exhibits no mesh tangling, ensuring mesh validity throughout;\n3. The adapted meshes generated show no significant element distortion, further guaranteeing mesh quality.\n\nWe also intended to include Columbia River and North Sea cases, but their bathymetry datasets were temporarily inaccessible. In the revised version, we will make every effort to include these cases and present their results in the supplementary materials.\n\n### **Response to Q4**\n\nThank you for your comments and feedback.\n\nFirst, we analyzed the reason for the high variance in the results on the Poisson equation. We have found that this discrepancy primarily stems from the fact that we did not update the mesh boundaries, while the distribution of those solutions leads to large Hessian values at the mesh boundaries. To illustrate this point, we allowed mesh nodes to move tangentially along the boundaries and retested the model's performance. The new experimental results show that when further permitting bound node movement, the model's performance improved significantly. \n\n| PDE    | Poisson1 | Poisson2 | Poisson3 | Poisson4 | Poisson5 | Poisson6 | Poisson7 |\n| - | - | - | - | - | - | - | - |\n| ER (%) | 17.49    | 14.82    | 13.08    | 15.98    | 11.18    | 17.85    | 15.69    |\n\nSecondly, we would like to point out that the r-adaptive mesh method struggles to maintain a consistent error reduction ratio across different problems due to significant variations in physical properties, error distributions, and length scales among different flow fields. Its effectiveness heavily depends on the local characteristics of the flow field and the accuracy of the error indicator. Moreover, constrained by the requirement to preserve mesh topology, it cannot adapt to more complex error structures by adjusting the number of mesh elements. **While the r-adaptation is effective, it faces challenges in ensuring consistent error reduction across varying flow fields.**\n\nBeyond highlighting this issue, we believe a more critical question is whether the method can reliably reduce errors (e.g., some approaches may degrade accuracy in certain scenarios), as this reflects the algorithm's robustness. To evaluate this, we tested the adaptation performance on 1,000 randomized cases (with random boundaries and flow fields). The results show that our method achieved effective accuracy improvement in 807 cases, demonstrating its robustness across diverse scenarios.\n\n### **Response to Q5**\n\nThe adapted mesh is computed from **a uniform mesh** at each step (the same as M2N and UM2N). At every time step, we project or interpolate the flow field from the previous time step onto the uniform mesh, then compute the current time step's mesh and use it as the input for the next step. \n\nAn autoregressive approach is promising, as it could avoid an interpolation step. However, since the input mesh would become more diverse, the model would need to handle a wider variety of mesh distributions—making it challenging to construct a dataset that covers sufficient cases. This issue is certainly worth further investigation in future research.\n\n### **Response to Q6**\n\nThank you for your suggestion. We have evaluated the performance of the benchmark methods on these two  cases, with the following results:\n\n1. MA: For both  cases, the MA method exhibited divergence behavior during the MA equation solving and failed to generate valid adapted meshes.\n2. M2N: In the moving cylinder case, the M2N did not demonstrate consistent behavior. While it could generate well-adapted meshes at certain time steps, it produced highly distorted mesh elements at others. For the supersonic case, the M2N could capture the shock position, but with lower mesh node density at the shock compared to UGM2N.\n3. UM2N: In the supersonic case, although the UM2N could identify the shock position, it introduced  element distortion in irrelevant regions. For the moving cylinder case, while it could fit the flow field distribution, excessive adjustments resulted in distorted mesh elements.\n\nThese comparisons further demonstrate the robustness of UGM2N under diverse flow conditions and complex boundaries. Additional results will be included in the revised supplementary materials.\n\n### **Response to Q7**\n\n**Efficiency Analysis**\n\nWe appreciate your reminder regarding the insufficient discussion on time efficiency. In Fig. 4, we present the mesh adaptation time (multiple epochs) of UGM2N under various mesh sizes, where the dashed line indicates the time for a single epoch. It can be observed that the single-epoch adaptation efficiency of our model is higher than that of the M2N and UM2N, allowing us to maintain good efficiency even under multi-epoch adaptation.\n\nTo further analyze the time allocation of different operations within a single round, we have constructed meshes with varying numbers of mesh elements to test model inference time (as \"infer. time\"), mesh node interpolation time (as \"inter. time\"), and I/O time, and the results are shown below:\n\n| Element number | Infer. time (ms) | Inter. time (ms) | IO. time (ms) |\n| - | - | - | - |\n| 1478           | 14.3±0.12       | 0.748±0.015     | 48.7±0.84    |\n| 2396           | 17.9±0.12       | 1.30±0.30      | 75.9±0.85    |\n| 4126           | 25.5±0.45       | 2.49±0.021      | 136±0.39     |\n| 9248           | 43.9±0.67       | 7.17±0.070      | 302±0.83     |\n| 36098          | 354±1.88         | 60.0±0.60       | 1010±1.55     |\n\nIt can be observed that, in a single adaptation epoch, the majority of the time is consumed by I/O operations (data transfer between CPU and GPU), while model inference time and interpolation time remain relatively low. The I/O time consists of two operations:\n\n1. Transferring mesh data (typically stored on the CPU) to the GPU before adaptive adaptation begins.\n2. Transferring the newly adapted mesh nodes from the GPU back to the CPU for interpolation.\n\nIf a GPU-based Delaunay triangulation interpolation function were available, the second data transfer could be avoided, significantly improving our model's inference efficiency. However, we have not found a mature GPU implementation of such a function, and developing one falls outside the scope of this work. In the future, we will focus on developing this functionality to further enhance the efficiency of our method.\n\n\n**Convergence**\n\nThis is an insightful and important observation.  We suggest that this phenomenon could potentially relate to data shifts occurring during adaptation. Our model was trained on uniform meshes, and  the distribution of adapted  mesh elements progressively deviates with increasing iterations. This distributional shift introduces **untrained mesh node distributions**, causing randomized model outputs that may subsequently lead to mesh distortion and eventual tangling.\n\nThe most straightforward  solution  is our dynamic termination strategy. This strategy automatically halts the adaptation process when the  $\\mathcal{L}_{\\mathrm{var}}\\left(\\mathcal{M}^{\\prime}\\right)$ ceases to decrease, thereby preventing further quality degradation. This solution effectively addresses the convergence challenges while maintaining computational efficiency.\n\n------\n\nThank you again for your careful review and valuable feedback. We hope our responses have addressed your concerns, and we welcome any additional questions or suggestions you may have."}, {"review_id": "wXoDMsNFxy", "reviewer": "Reviewer_dnRz", "summary": "The 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.", "questions": "N/A", "limitations": "Yes", "rating": 3, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 1, "strengths_and_weaknesses": "Novelty: 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.", "was_revised": true, "final_justification": "The authors have addressed my concerns on originality and time complexity. I update my too low score but remain borderline reject due to the limited extent of the novelty on the UM2N baseline.", "rebuttal": "### **Response to Concerns Regarding Novelty and Originality**\n\nThank you for your detailed evaluation and thoughtful comments on our manuscript. We appreciate your concerns regarding the novelty of our contributions and would like to clarify the originality of our approach through a point-by-point response.\n\nFirst, we acknowledge the relevance of the recent work by Rowbottom et al. and appreciate the opportunity to clarify the citation timeline. Our research began in September 2024, during which the latest available version of their paper was the July 2024 preprint, which we cited. At that time, the preprint did not include elements overlapping with our locally-developed M-Uniform loss function.  The ICML 2025 version was announced as accepted in May 2025 (the final published version will be available even later), and we only became aware of its final content after submitting our manuscript. Since then, we have carefully reviewed the latest version of Rowbottom et al.'s work.  Rowbottom et al.'s work represents a significant and timely contribution to the field, demonstrating impressive theoretical depth and empirical rigor. Their integration of GNNs with Firedrake to minimize finite element solution errors and the Graph DiffFormer architecture represent a notable contribution, particularly in addressing mesh entanglement.  We will ensure proper citation  of this work in our revised manuscript.\n\nSecond, our two key innovations - the M-Uniform Loss and node-patch based mesh adaptation approach - remain fundamentally distinct from Rowbottom et al.'s work. As you noted, our method focuses entirely on local node optimization, while they continue using M2N/UM2N's whole-mesh input paradigm. Although their the ICML version's Section 5.6 mentions patch-based processing, two crucial differences exist: (1) Their approach resembles mesh partitioning rather than our fine-grained node-level processing, which works better for complex geometries where partition may vary significantly; (2) Their patch-based method was actually introduced during the review process (see their rebuttal  on OpenReview, which is visible to everyone only after June 18, 2025 ) and was absent from their February 2025 version. This timeline strongly  confirms the originality of our approach.\n\nFinally, we would like to highlight additional contributions of our work, including: (1) a fully label-free training approach that relies solely on monitor function values, in contrast to methods requiring FEM gradient computations coupled with Firedrake; (2) a mesh-smoothing-inspired iterative optimization scheme that demonstrates consistent improvement across a variety of problem settings; and (3) zero-shot generalization capability, with our model trained on a limited dataset achieving effective performance across multiple PDE cases, compared to approaches that often require case-specific training.\n\nWe hope these explanations will help reassess our work's novelty and originality. Below we address your other constructive comments point-by-point (marked as **Q** for questions).\n\n### **Response to Q1**\n\nWe appreciate the clarification request. As described in Section 3.2 (lines 200–205), our training data consists of pre-generated local node patches extracted from high-quality reference meshes (see Figure 10). During training, these patches are treated as static samples—there is no dynamic mesh update or policy rollout. Therefore, the training distribution remains stable, and there is no distribution shift between epochs.\n\n### **Response to Q2**\n\nIn fact, in Figure 4 we have presented the time efficiency of different mesh adaptation methods across various mesh sizes, where our method's timing already includes interpolation operations. The results in Figure 4 demonstrate that our interpolation process does not introduce significant computational overhead, enabling our method to maintain highly efficient single-iteration performance while also achieving substantial reductions in multi-iteration adaptation time.\n\nTo provide further evidence, we have conducted timing experiments across different mesh sizes. The results, presented below, show that the main bottleneck is I/O between CPU and GPU, while both inference and interpolation times remain low. These findings confirm the scalability of our method, and we plan to further optimize I/O bottlenecks in future work.\n\n| Element number | Infer. time (mean±std ms) | Inter. time (mean±std ms) | IO. time (mean±std ms) |\n| -------------- | ------------------------- | ------------------------- | ---------------------- |\n| 1478           | 14.3±0.119                | 0.748±0.0154              | 48.7±0.839             |\n| 2396           | 17.9±0.116                | 1.30±0.301                | 75.9±0.853             |\n| 4126           | 25.5±0.448                | 2.49±0.0211               | 136±0.387              |\n| 9248           | 43.9±0.669                | 7.17±0.0703               | 302±0.832              |\n| 36098          | 354±1.88                  | 60.0±0.603                | 1010±1.55              |\n\n### **Response to Q3**\n\nThank you for your valuable feedback. Although we have not theoretically analyzed how to guarantee error reduction without mesh tangling, we have demonstrated the excellent performance of our model in error reduction and avoiding mesh tangling through extensive and diverse experiments. Specifically, we have verified the model's performance under varying flow field solutions, mesh resolutions, and mesh shapes, all of which confirm that our method achieves improved solution accuracy without any observed mesh tangling. Overall, our proposed method exhibits stronger robustness and generalization compared to existing traditional methods and ML-based mesh adaptation approaches.\n\nTo further illustrate the generalization capability of our model across different scenarios, we have added the following new experiments:\n\n1. **Random Geometry Test**: On 1,000 randomly generated meshes (with 3–6 edge polygons and complex flow fields), our method improved solution accuracy in 807 cases — the highest among all baselines.\n2. **Real-world Case**: On a challenging tsunami simulation case with complex coastal boundaries, our model successfully adapted the mesh without producing any invalid elements.\n3. **Non-convex Test**: On 100 non-convex domains prone to mesh entanglement, our method achieved the best error reduction performance and produced no tangled meshes.\n\nThese results indicate that our model offers not only strong generalization but also practical robustness in avoiding mesh tangling under a wide range of conditions.\n\n------\n\nWe are truly grateful for your detailed and thoughtful comments. They have given us a clearer perspective on both the strengths and limitations of our work. We look forward to your follow-up and are happy to provide further clarification if needed."}, {"review_id": "GyLiNomVvC", "reviewer": "Reviewer_YRmJ", "summary": "This 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.", "questions": "**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?", "limitations": "1. lack of real-world scenarios benchmarks.\n2. the writing of this paper is hard to follow.", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "Strenth:\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.", "was_revised": false, "final_justification": "", "rebuttal": "We sincerely thank the reviewer YRmJ for their valuable comments and suggestions, which help significantly improve the quality of this manuscript. Below, we provide a point-by-point response to your feedback, with annotations indicating whether each item pertains to a **Question (Q)** , **Weakness (W)** , or **Limitation (L)** in our original submission.\n\n### **Response to W1\\L1**\n\n**Writing and Typo Issues.**\n\nWe sincerely appreciate your feedback. In the expression $σ =∫_K ρ(x) dx$, the function $\\rho(x)$ should actually be $m(x)$. We apologize for this notational error in our manuscript. For the clarity of our writing，we recognize that certain sections may have been overly technical or insufficiently structured, which could hinder readability. To address this concern, we will take the following steps in revision:\n\n1. **Improved Organization** – We will refine the logical flow of ideas, ensuring a clearer progression from motivation to methodology and results.\n2. **Enhanced Explanations** – Key concepts and assumptions will be expanded with additional intuitive examples or illustrative diagrams where appropriate.\n3. **Language Edits** – Complex sentences will be simplified, and terminology will be consistently defined to improve accessibility without sacrificing rigor.\n\nWe are committed to making the paper more accessible to a broader audience while maintaining its technical depth. Please let us know if specific sections require further attention, and we would be glad to provide additional clarifications.\n\n### **Response to W2**\n\n**Justification of the assumption.**\n\nThank you for raising this insightful question.  In our M-Uniform loss, we introduced $Var(L_K) ≈ Var(L'_K)$ as a mild assumption. Since our goal is to handle triangular meshes, we also assume that our meshes possess a certain level of quality, which is achievable with modern mesh generation software—even for highly irregular or complex geometries.\n\nIn fact, our requirement that *each mesh element appears in a similar number of local patches* essentially means that each mesh node should have a similar degree (i.e., the number of connections to neighboring nodes). Generally, boundary nodes are relatively few compared to interior nodes, and most interior nodes have a degree of 6 or 5 (with a small fraction being 4, which already indicates a more distorted element). In a 2D test case with 60K mesh elements generated by mature mesh generation software, we observed that 92.96% of nodes had a degree of 6, while 2.96% had a degree of 5—meaning around 95% of nodes fell into these two categories. Therefore, we consider this a reasonable assumption.\n\nTo further demonstrate our model's performance when this assumption does not hold, we have tested it on real-world cases with arbitrary geometry boundaries, specifically in the Tohoku tsunami simulation (ocean modeling), where only about 69% of nodes had degrees of 6 or 5. The results showed that our model could still accurately capture wave propagation, with mesh element sizes dynamically adapting to wave propagation. This also highlights our model's generalization capability for highly irregular or complex geometries. Additionally, we have evaluated our model's performance on highly irregular meshes where mesh elements do not appear in a similar number of local patches. The flow field solution for each sample was defined as $\\cos(2  \\pi  (x - \\mu_x))$, where $\\mu_x$ is a random value in [0, 1]. The error reduction results achieved by our model are presented below:\n\n|            | Case 1 | Case 2 | Case 3 | Case 4 | Case 5 | Case 6 | Case 7 | Case 8 | Case 9 | Case 10 |\n| ---------- | ------ | ------ | ------ | ------ | ------ | ------ | ------ | ------ | ------ | ------- |\n| **ER (%)** | -2.07  | 16.07  | 10.69  | 11.11  | 18.10  | 14.24  | -2.10  | 8.46   | 9.84   | 7.16    |\n\nThe experimental results demonstrate that even when mesh elements do not appear in a similar number of local patches, our model can still achieve accuracy improvement in most cases(achieving success in 8 out of 10 cases).\n\n### **Response to W3\\L2**\n\n**Real-world scenarios.** Thank you for your suggestions. Based on your feedback and Reviewer AXEC’s comments, we have added a **real-world case with arbitrary geometry boundaries**, sourced from Thetis project in ocean modeling. The experimental results demonstrate that our model dynamically adapts mesh elements to accurately simulate wave propagation while maintaining high-quality, distortion-free meshes without tangling, even under complex boundary conditions. \n\nDue to technical constraints (e.g., one case requiring additional data access and another involving website maintenance), we were unable to include the other two real-world scenarios (**Columbia River plume simulation and North Sea tidal simulation**) at this stage. However, we will make every effort to incorporate all three cases in the revised manuscript to further validate the robustness of our method.\n\n**Validation in 3D large-scale scenarios** \n\nThank you for your feedback. Theoretically, our method imposes no restrictions on mesh scale, as our GNN-based model can accept graphs of arbitrary size as input. However, in practice, this often leads to exceeding GPU memory limits.\n\nOn a single RTX TITAN GPU with 24GB memory, our model can handle approximately 40,000 mesh elements, with model inference taking about 345ms and interpolation requiring around 61ms per epoch. In comparison, the MA method takes approximately 25510ms, demonstrating that our approach offers significant efficiency advantages over the MA method. We have also verified the proportion of model inference recommendation, interpolation time, and I/O time under different numbers of mesh elements, and the results are as follows. It can be seen that I/O time is the main performance bottleneck, which inspired our future optimization direction.\n\n| Element number | Infer. time (ms) | Inter. time (ms) | IO. time (ms) |\n| - | - | - | - |\n| 1478           | 14.3±0.12        | 0.748±0.015      | 48.7±0.84     |\n| 2396           | 17.9±0.12        | 1.30±0.30        | 75.9±0.85     |\n| 4126           | 25.5±0.45        | 2.49±0.021       | 136±0.39      |\n| 9248           | 43.9±0.67        | 7.17±0.070       | 302±0.83      |\n| 36098          | 354±1.88         | 60.0±0.60        | 1010±1.55     |\n\nOn the other hand, our method is highly suitable for parallelization because there are *no data dependencies* between individual node patches. This allows for easy distribution across N GPUs to enable large-scale mesh processing. Additionally, we can trade time for space by processing patches of mesh points incrementally, thereby increasing the manageable data scale. Adopting this approach (without considering additional overheads like patch grouping and aggregation), our model's processing time for meshes on the order of 1 million elements is roughly 10s, which still outperforms the MA method (which already takes 25s for 40,000 elements).\n\n### **Response to Q**\n\n**Geometric rotation invariance property.**\n\nWe sincerely appreciate the reviewer for raising this intriguing question. First, we attribute these minor discrepancies to inherent asymmetries in the underlying mesh triangulation—rotating the test mesh by 90° fails to achieve perfect overlap with the original mesh. More importantly, this observation prompts a fundamental investigation into whether intelligent mesh adaptation methods possess rotation invariance: does rotating the mesh (while preserving nodal flow variables) affect error reduction performance?\n\nIn principle, arbitrary angular rotations should not influence error reduction when flow field variables remain unchanged at mesh nodes. However, the black-box nature of neural networks makes this difficult to guarantee theoretically. To empirically validate this property, we have conducted systematic experiments rotating both the Helmholtz case mesh and its flow field through 360° (in 10° increments), evaluating error reduction across all angles for different ground truth solutions. The comprehensive results are presented below:\n\n| PDE Type   | MA               | M2N             | UM2N         | UGM2N        |\n| - | - | - | - | - |\n| Helmholtz1 | 15.60 ± 0.0001   | -16.42 ± 16.54  | 12.03 ± 0.76 | 12.25 ± 4.32 |\n| Helmholtz2 | 10.30 ± 0.0079   | -27.72 ± 10.02  | 8.16 ± 1.45  | 10.64 ± 3.79 |\n| Helmholtz3 | 13.48 ± 0.0000 | -17.54 ± 7.75   | 6.58 ± 1.08  | 12.87 ± 3.38 |\n| Helmholtz4 | 10.87 ± 0.0000 | -282.57 ± 49.40 | -0.92 ± 1.56 | 11.92 ± 1.86 |\n| Helmholtz5 | 13.50 ± 0.0000 | -242.00 ± 64.94 | 4.67 ± 1.66  | 14.33 ± 2.34 |\n\nThe results clearly demonstrate that, compared to the MA method, intelligent mesh adaptation approaches exhibit poorer rotation invariance, as evidenced by their larger performance variance ranges. This insight suggests a valuable direction for future research to enhance geometric invariance in learning-based adaptation methods.\n\nWhile our method shows inferior rotation invariance compared to UM2N, we emphasize that our node-patch processing framework inherently possesses two critical advantages:\n\n1. **Scale invariance**: The adaptation results remain unchanged under uniform scaling of the mesh.\n2. **Translation invariance**: The output are consistent  under arbitrary translational transformations of the grid.\n\nThese properties, which M2N and UM2N fundamentally lack, provide significant practical benefits in real-world applications where mesh scales and positions may vary. The limitation in rotation invariance remains an open challenge that requires further investigation in the future work.\n\n------\n\nThank you very much for your constructive and insightful feedback. We have carefully considered each point and provided detailed responses. We welcome any further comments and look forward to the next stage of the review process."}]

Author rebuttal:

Meta-review: The authors proposed an unsupervised mesh adaptation approach for numerical simulations. To process the whole mesh the method considers local mesh neighnorhoods. To train the model for mesh adaptation the approach uses a physics-constrained loss function to enforce mesh equidistribution. The authors demonstrated that the method works across different PDEs and mesh geometries and can significantly improve accuracy of simulations. 

As a strengths of the paper I would highlight 
- the paper is rather clearly written and properly motivated
- the method is unsupervised, so as a result the method does not require to generate high-quality adapted mesh
- the experimental section convincingly demonstrates that the proposed approach outperforms other correcting schemes
-  the model demonstrates generalization capability across multiple test cases, enabled by its few-shot learning ability, so it eliminates the need to prepare large-scale training datasets

As a weaknesses I would highlight
- the text should be further polished. E.g. there are many specialised terms in the text that could be not so familiar for DL/ML community
- there is no theoretical analysis confirming convergence or approximation guarantees
- the method currently does not have specific mechanisms to work with boundary nodes, which could be crucial for some real-world problems

The main reasons for the accept decision are
- unsupervised approach for mesh adaptation with good generalization capabilities across multiple test cases
- the approach is rather computationally efficient
- extensive empirical evaluation

During the discussion with the reviewers several issues were raised that should be addressed in the final version of the paper. Here are some issues
- reviewer mhZx proposed to include explanations of some specific terms from computationsl physics such as GAT, Monge-Ampère based method, etc.
- reviewer Htea proposed to consider boundary limitation issues of the proposed method
- the authors did a lot of additional experiments in response to authors questions. These results should be included in the main text and in the appendix
Decision: Accept (poster)