M2N: Mesh Movement Networks for PDE Solvers
Authors: Wenbin Song
Mingrui Zhang
Joseph Gregory Wallwork
Junpeng Gao
Zheng Tian
Fanglei Sun
Matthew D Piggott
Junqing Chen
Zuoqiang Shi
Xiang Chen
Jun Wang
Venue: NeurIPS 2022 Accept (2022)
Keywords: Partial Differential Equation
Mesh Adaptation
Mesh Movement
Moving Mesh
r-Adaptation
Monge–Ampere
Deep Learning
Neural Network
Neural Spline
Graph Attention Network

Abstract: Numerical Partial Differential Equation (PDE) solvers often require discretizing the physical domain by using a mesh. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without introducing extra computational burden to the PDE solver, by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, sophisticated mesh movement methods, such as the Monge-Ampère method, generally require the solution of auxiliary equations. These solutions can be extremely expensive to compute when the mesh needs to be adapted frequently. In this paper, we propose to the best of our knowledge the first learning-based end-to-end mesh movement framework for PDE solvers. Key requirements of learning-based mesh movement methods are: alleviating mesh tangling, boundary consistency, and generalization to mesh with different resolutions. To achieve these goals, we introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline based model provides more flexibility for large mesh deformation, the GAT based model can handle domains with more complicated shapes and is better at performing delicate local deformation. We validate our methods on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampère method, our approach can greatly accelerate the mesh adaptation process by three to four orders of magnitude, whilst achieving comparable numerical error reduction.

M2N: Mesh Movement Networks for PDE Solvers
Wenbin Song ShanghaiTech University Saagsb@shanghaiteechh edu. cn
Mingrui Zhang Imperial College London mingrui. chang18@imperian ac. uk
Joseph G. Wallwork Imperial College London
Junpeng Gao
ETH Zürich jungao@student. ethz. .h
wallwork16@imperiaan ac. uk
Zheng Tian
Fanglei Sun
Shanghai University zheng. tian. wewucc.eac......
ShanghaiTech University sunf Sunf@@@hanghaitere edu cn
Matthew D. Piggott Imperial College London
Junqing Chen
Tsinghua University j jgchen@tsinghua. edu.
m. d. piggottsiimpeiial.eee ac.uu.
Zuoqiang Shi Tsinghua University
Xiang Chen
Jun Wang
Noah's Ark Lab, Huawei xi xiangchen.c. ei@utloooo.... com
University College London
zqshi grehiStsinghua. edu. en
jun wang@cs ucl ac
Abstract
Numerical Partial Differential Equation (PDE) solvers often require discretizing the physical domain by using a mesh. Mesh movement methods provide the capability to improve the accuracy of the numerical solution without introducing extra computational burden the PDE solver, by increasing mesh resolution where the solution is not well-resolved, whilst reducing unnecessary resolution elsewhere. However, sophisticated mesh movement methods, such as the Monge-Amper method, generally require the solution of auxiliary equations. These solutions can extremely expensive to compute when the mesh needs to be adapted frequently. In this paper, propose to the best of our knowledge the first (earning-bascd end-to-end mesh movement framework for PDE solvers. Key requirements alleviating mesh tangling, boundary
learning- baased mesh movement methods consistency, and generalization to mesh with different resolutions. To achieve these goals, introduce the neural spline model and the graph attention network (GAT) into our models respectively. While the Neural-Spline hased model provides flexibility for large mesh deformation, the GAT' based model handle domains with more complicated shapes and is better at performing delicate local deformation. We validate methods stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains. Compared to the traditional Monge-Ampère method, our approach can greatly accelerate the mesh adaptation process by three to four orders of magnitude, whilst achieving comparable numerical error reduction.
"Equal contribution.
Corresponding Authors: Xiang Chen and Jun Wang.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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1 Introduction
Partial Differential Equations (PDEs) are widely used to model natural phenomena, ranging from astrophysics and ocean dynamics to semiconductor device simulation and bio-engineering [Evans] 2010]. Acquiring accurate numerical solutions efficiently for complex PDEs is an essential but challenging problem in all scientific and engineering disciplines. Solving PDEs using numerical methods such as the Finite Element Method (FEM) requires discretizing the problem spatially and temporally. A mesh is often used for spatial discretization and its quality affects the accuracy of the numerical solution [Frey and George 2007]. However, it is often prohibitively expensive to solve the problem on very high resolution mesh. Mesh adaptation is an advanced discretization method designed to tackle this problem. It increases the mesh resolution where the solution requires higher numerical accuracy, while decreasing the mesh resolution where unnecessary. Mesh adaptation methods can be generally divided into two categories: h--daptaaion and r-adaptation. In h-adaptation, new mesh nodes are dynamically added to the regions where fine resolution is required. In r- adaptation (or mesh movement), however, mesh nodes are only relocated or moved without changing the mesh topology (Huang and Russell 2011]. Compared to h-adaptation, r-adaptation has several attractive features. First, no extra mesh points are generated, which keeps the dimension of the linear system representing the discretized PDE unchanged. In addition, fixed mesh connectivity can also make the structure of the stiffness matrix unchanged, which enables matrix pre- pre-factorization to accelcrate the solution of the large linear systems encountered in FEM [Budd et al., 2009]. However, common problem of mesh movement methods is mesh tangling. in which lines connecting the mesh nodes come across each other. Mesh movement methods based on optimal transport theory can effectively prevent mesh tangling issues (see [Clare et al. 2022] for a discussion on this), but require solving a Monge-Amper equation at each adaptation step, which is highly time- inefifeefiii
Al-powered approaches to computing solutions of PDEs have been emerging topic in recent years, and show great potential in solving problems where traditional numerical PDE solvers struggle (e.g. high dimensional problems [Han et al. 2018 (heng and Yang 202T)), or in accelerating the solution process by learning a neural operator from the parameterized description of PDE problem to its corresponding solution (Li et al. 2020 Lu et al. 2019]. However, these methods still encounter fatal bottlenecks, such as the precision guarantee of the solution, data efficiency, generalization capability, etc. These fundamental limits of deep learning, but essential for scientific computing scenarios.
possible alternative is to perform learning- based mesh adaptation, by which the traditional numerical PDE solvers can achieve better performance, while the time consumption of mesh adaptation is greatly reduced. There have been some works in this direction [Zhang et al. 2020 Yang et al. 2021
Huang et al.] 2021 Fidkowski and Chen 2021 Tingfan et al. 202 20.1 Pfaff et al. 2020]. However, most previous methods focus on mesh generation /Zhang et al. 2020) or mesh refinement (Yagg ct al. 2021 Huang et al. 202 Fidkowski and Chen 2021]. instead of copology-invarian
mesh movement as considered in this work. Moreover, the prior works are not end-to-end approaches, which means the neural networks are used to predict certain metrics, such the local mesh density (Zhang et al. 2020 Huang et al. 2021] or the metric tensor [Fidkowski and Chen 2021 Tingfan let al., 2021 Pfaff et al.) 2020), which then have to be fed into a traditional mesher/remesher to obtain the mesh. Therefore, the overall performance is bounded by the mesher/remeseer, which is computational geometry based and hence not optimized for solving PDE problems. On the contrary, in our method, the adapted mesh is directly output by the neural network.
In this work, we propose to the best of our knowledge the first learning- based end-to-end mesh movement framework for PDE solvers. Taking the source term, the input field, and/or the PDE parameters input features, the model deforms an initial mesh to the adapted mesh by mesh movement. In usage, the model can be applicd to a class of PDEs without retraining. We design a Neural-Spline based model for mesh deformation. It is an invertible neural network and hence can avoid mesh tangling. Moreover, its mechanism naturally guarantees that hypercubic boundary can be maintained through learnable mapping. We also design a graph attention network (GAT) based model for mesh deformation. The graph neural network can naturally describe domains with irregular shapes and embed the relevant information. We utilize the attention mechanism of the GAT model to guarantee each mesh node stays within its neighborhood so that mesh tangling can be alleviated.
Our main contributions are listed as follows:
2

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1. We propose Cearning-based end-to-end mesh movement framework for PDE solvers, which to the best of our knowledge is the first of its kind. Without interfering with the PDE solver, the models can achieve numerical error reduction similar to the traditional Monge-Ampère method, while the mesh generation is accelerated by three to four orders of magnitude.
2. A Neural-Spline based model and GAT based model are proposed for mesh deformation. Besides generalization to different PDE parameters, source terms, input solution fields, etc., the models are designed to guarantee boundary consistency, alleviate mesh tangling, and generalize to different mesh densities, which are all desired for mesh movement.
Related Work
Mesh movement method. Mesh movement methods include velocity- -based and location based
methods. In the work of [Anderson and Rai 1983 Gnoffo
1982 Farhat et al.) 1998]. the mesh is
moved according to attraction and repulsion pseudo-forces between nodes motivated by spring model. The moving mesh finite element method [Baines et al. 2005 computes the solution and the mesh simultaneously by minimizing the residual of the PDHs written in a finite element form. As for location-bac based method, the moving mesh PDE (MMPDE) method (Huang and Russell [201]] moves the mesh through the gradient flow equation of an adaptation functional. In recent years, there has been a growing interest in ppiimally-trnnsoortee r-adapted meshes [Budd and Williams, 2009 Clare
et al. 2022
AI for PDE. To solve PDE problem, neural networks can be used to represent the function to solve, and trained either with the residual loss of the PDE or using the variational principle [Raissi al. 2019 Yu al. 2017|. Neural operators are proposed learn an operator from the problem function to the solution function Li et al., 2020 Lu el al. 2019]. There also exist mesh-based PDE solvers with deep learning. In (Pfaff et al., 2020), graph neural network with additional world edges is applied to predict dynamical systems, shown to be effective with wide range of physical systems. In Selbutee-Peres al.) 2020], differentiable PDE solver is embedded in a neural network to help predict accurate solutions and also backpropagate the loss so that the input coarse mesh can be optimized.
AT for Meshing. Al methods have also been proposed for mesh generation, adaptation, and so on. MeshingNet (Zhang al., 2020] uses neural network learn the required local mesh density, which can then be provided to Delaunay triangulation based mesh generator to generate high-quality meshes. The optimal local mesh density is also learned in (Huang et al. 2021] for mesh refinement. On the other hand, the mesh refinement process is formulated as reinforcement learning problem in Yangg al. 2021] to minimize the PDE solution error under given refinement budgets. The flow field is predicted by machine learning models to calculate the metric tensor so that the mesh can be optimized accordingly [Tingfan al. 2021] The authors (Fidkowski and Chen 2021 focused on optimal anisotropic meshes by predicting the desired element aspect ratio. In [Pfaft et al. 2020). the sizing field is predicted by a neural network for adaptive remeshing along with the system dynamics.
Method
3.1
Problem Statement
For each learning task, we consider a class of PDE problems P defined on the domain Ω in D dimen- sions. Each specific sample P, E IP (n 1, 2, is determined by PDE related information, such as source terms, boundary conditions, PDE parameters, solution fields, etc. We use X,, to param- eterize such information, which will be fed into the neural network as raw features, so that the trained model can generalize to a class of PDEs. The sample-spechos further categorized into global parameters P.. and position-dependeer parameters M,, Depending on the type of the PDE problem and the terms we expect the trained model to be able to generalize to, some examples of P., are wave number of the Helmholtz equation and viscosity coefficient of the Burgers' equation, and options of M,, are source terms, system states in the previous timestep for time-dependent equations, etc.
information X', = {p.,. M, can be
3

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(a) Framework Overview
(b) Movement Network Implementation
Preblan:2
Solution ul!
Global Parameters Ps
2 PDE Solver
Ma
Neural
Global Festure
Initaal Mosh
Adopted Mush MIN
Medh Movement Natwork Fazz
M2N Splinne
M2N Framework
Glotal Festurre
(é) Global Feature Extractor
Graph
GNN Block
GAT
Constrictor
Sample
Cocov
M2N GAT
Global Parameters Pa
Figure 1: Proposed mesh movement network framework. (a) Given an initial mesh and an input state, the mesh deformer outputs an adapted mesh, which is then fed to the PDE solver. (b) The implementations of the Neural-Spline and the GAT based mesh movement networks. (c) The structure of the Global Feature Extractor.
Meshing is the procedure to spatially discretize a domain, which is necessary for most numerical methods to solve PDEs. Among these methods, the Finite Element Method (FEM) has been widely used in various engineering fields, which we will take as our solving method in the experiments. The initial mesh is generated with traditional mesh generator. We denote the initial mesh by T and the mesh node positions in Tini by ginit. Since the generation of the initial mesh is purely computational geometry based, the sample-specifis information X,, cannot be utilized, hence Tini is shared by all samples P. P. A standard PDE solver S will take as input the PDE problem Pn, discretized on the mesh Timit and output the corresponding solution u = ((PP((iii
A high-quality adapted mesh can significantly improve the securacy-effiiiene trade-off of the PDE numerical solution. There are in general two types of mesh adaptation techniques. In this work, we consider mesh movement, r-adaptation, which means we will not modify the number and topology of mesh nodes, but only relocate them. This characteristic provides the advantage that no remeshing needs to be performed, and there will be less repetitive computation for the PDE solver since the input matrices will maintain a constant size and sparsity structure [Budd et al. 2009]. A typical mesh movement method F will map the nodes ginit in the initial mesh T to C.n = =F(g inii Xn),,
so that the adapted mesh T will be constructed with gulpp and the topology defined on the initial mesh. The PDE solution corresponding to the adapted mesh discretization ut, uh S(P, (Ta.le)) is expected to be much more accurate than init although the scale of the discretized problem that solver S receives is exactly the same. One of the most advanced mesh movement methods is the Monge-Ampère (MA) method. The mesh nodes optimized by the MA method eMA. (e Xx,) will be taken as the supervised signal to train the neural network model, while the computational efficiency and accuracy of the corresponding solution M. (TMMA)))) will be compared against.
3.2 Framework Overview
The computational cost of sophisticated traditional mesh movement methods is often too expensive. In some cases, the cost of mesh adaptation is comparable to or even higher than that of solving the underlying PDE problem, which generally unacceptable. Therefore, our goal is to model a Mesh Movement Network (M2N) based mapping F-))(-10), where 0 represents the trainable parameters, such that the mesh adaptation process can be greatly accelerated. The proposed framework is demonstrated in Figure [ We consider constructing a learning-base based mesh movement method that relocates the nodes iii in the initial mesh given the input state X
<n
M2N = FwaN(E init x,,)),
(1)
from which the adapted mesh M2N can be reconstructed based on the topology of the initial mesh. From empirical study, take el loss between the model output and the adapted mesh nodes

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obtained with the Mlonge-Ampers method:
L(
Σ P.
Sn
M2N
eMA 57
(2)
Under the M2N framework, we implemented two network structures: a Neural-Spline based network and a GAT based network, which are denoted as M2N-Spline and M2N-GAT, respectively. Both models are designed to be capable of alleviating mesh tangling, keeping boundary consistency, and generalizing to mesh with different resolutions, which are key requirements for mesh movement. Because of their inherent characteristics, the M2N-Spline model behaves better when large global mesh deformation is required, while M2N-GAT able to handle irregularly shaped domains and can better learn delicate local deformation. The detailed model structures will be introduced in the following sections, and more specifics of the model implementations can be found Appendix A.
3.3 Neural-Spline based Network
As demonstrated in Figure bb), our Neural-Spline based network is mainly composed of two parts, where the input information Xr, = (P.,:MM will be separately fed into the model. The global feature extractor GFB(-) extracts features from the position-dependent parameters Mn to obtain the mesh resolution invariant embedding E,, E, together with the global physical parameters p,, will then be fed into the neural spline deformer control the mesh node relocation, i.e.
Splino
xx,) Spline GFF(MM) ( P.):
where operator @ represents tensor concatenation.
Global Feature Extractor
As shown in Figure cc), the global feature extractor has three modules:
GFE(M,)) = (AA(Con((Sample(A (MM,)))
(4)
In detail, uniformly sample the input state M,, in the domain Ω. If the domain is with irregular boundary, sampling is performed inside its minimum bounding box and the values the sampling points outside the domain boundary are set zero. The sampled states are assembled as a state tensor M,. To keep the extracted feature invariant to the magnitude, the state tensor M,, is normalized by its maximum absolute value. After normalization, M, sent into the convolutional layers for feature extraction, whose output is further fed into a Global Average Pooling (GAP) layer [Lin et al., 2013) to obtain mesh resolution invariant global embedding E,. The embedding E,, will then be concatenated with global physical parameters P.. to obtain I,,, which is fed into the deformer.
Neural-Spline based Deformer Normalizing flow models [ooyyevv et al. 2020] are proposed to learn invertible mappings. Neural spline [Durkan et al., 2019]. as a specific type of normaliz- ing flows, transforms the input with a differentiable monotone rational- spline function RQ)(-IK) where K represents the learnable anchor points that determines the invertible map- ping. In model, the neural spline deformer Spline(5 In) is stack of neural spline layers (E a (()))))==)) each of which transforms one dimension of the input node coordinates g(d)) whose anchor points K are parameterized by the input features I, and the other dimensions of node coordinates {-<d)
Since the neural spline model is guaranteed to be an invertible mapping, mesh tangling can be alleviated. A specialty of the neural spline model that, the end points of the spline function are fixed, therefore the intervals of the input and the output can be kept unchanged. In case, this property utilized to preserve the mesh with a hypercubic boundary (c.g. a rectangle in the 2-D case). Moreover, since the neural spline learns continuous mapping, it can naturally generalize to different mesh resolution input.
3.4 GAT based Network
Although the Neural-Spline based network is well-designed for meshes of hypercubic domains, it is difficult to extend to domains with more general boundaries. Therefore, we also propose Graph Neural Network (GNN) based model, which can naturally handle irregular domains. As shown in
5

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Figure [T) the model consists of a two- wwo-branch feature extractor and a Graph Attention Network (GAT, [Veličković et al. 2017)) based mesh deformer GAT this subsection, we omit the sample index 7 when there is no misunderstanding, hence: M (5 x) = GAT LFE(MM, GFE(M) @ p).
Feature Extractor The feature extractor of the GAT- GAT-based network consists of two branches. One the global feature extractor same as Eq. (4. the output of which will also be concatenated with global physical parameters p to obtain I. The other is a GNN-based local feature extractor LFE(-). Our experiments empirically show that both branches are necessary for end-to-end performance.
For the local feature extractor, we need to first construct the input graph G = (V, E) for each sample. The graph shares the same node number and graph topology with the initial mesh T For each graph node index E {1...., [.]. the input feature v. is the sampling of the input state M at </Tht the position of node i in the initial mesh T To preserve the mesh density information, the node distance l|st - -5j are encoded into edge features e, as the relative edge features. The constructed graph will then be processed by the GNN block with message passing mechanism to propagate the local physical information across the graph. The edge features are updated by a Multi-LLayer Perceptron (MLP) fx()): e + fx (k-1)
2) ( and the node features are updated by summing up the surrounding edge features (( + EjENN e, where k E {1.2.... KGNN} means the layer index, and (ii) refers to the neighbors of node i. At the output of the final layer, the features of each graph node concatenated with the extracted global feature I to assemble H; vf (KeNn) + I. We use H = HI,.... H [i] to represent the extracted features of the entire graph.
(k-1)
V
GAT-based Deformer As shown in Figure the deformer consists of KGAT GAT blocks. The k-th block
GAT Deformer
takes two inputs, mesh node positions
ginit
{(0)
g())
((2)
g(k-1) and extracted features (k-1))
GAT
TQAT
where (k-1)) H (k-1)
+ (k--1)
H
SELU
SELU
1 H(
B
H2)
For the first block, H (0) H and (0)) = - The GAT block performs the following transforms:
Figure 2: The implementation of the GAT Deformer.
(b) H
SELU
Σ JEN.
oig (()) (k-1)) (((-1)
(1)
(k)
Σ JEN,
(K) (k-1) oi E
(5)
where (A) is the attention score indicating the importance of node j to node in the k-thh block, acquired by the method introduced in (Velikkoić et al. 2017], and (()) is learnable weight matrix. To keep the mesh shape consistent before and after deformation, the nodes originally on the boundary are restricted to move along the boundary of the domain An. The adapted mesh G will
be constructed from the output mesh node positions of the last block g ((Kaxr) Because of the attention mechanism, during the mesh vertex relocation process, the movement of each mesh node confined inside the convex hull composed of its 1-ring neighbors, hence effectively alleviating mesh tangling.
4 Experiment
We evaluate our proposed models against two learning- based baseline models and the traditional Monge-Ampérs (MA) method, using error reduction ratio compared to the numerical solution the initial mesh, mesh adaptation time consumption, and element inversion ratio, to validate their performance, generalization capability, and robustness. The experiments are conducted the stationary and linear Poisson's equation in both a square domain and an irregular heptagonal domain, and the dime-dependent non-linear Burgers' equation, where the supervised optimized meshes generated with the traditional MA method. Each experiment is run three times with different random seeds ensure the reliability of the model performances and provide mean and standard deviation of the results, which are summarized in the tables. More details of the experimental setup, dataset generation, model training, and experimental results can be found in the Appendix.

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Table Performance summary of the Poisson's equation problem on the square domain.
Method
Error Reduction (%)
Time (ms)
Element Inversion (%)
MA (traditional)
23.11
5220.99
0.00
M2N-Spline
20.82 + 0.35
5.55 + 0.01
0.00
HLP--Deform- Clip
16.74 + 0.90
3.02 0.03
1,60
M2N-GAT
20.38 = 0.51
9.09 - 0.02
0.00
GAT-Deform-Clip
19.95 0.38
10.41 0.05
3.11
As mentioned in Section | there is
similar previous work to compare
results against. Comparisons with
learning-based I--ddattttion [Pfaff
et al., [202 ZZhang et al. 2020 Fid- kowski and Chen [20e]] Huang et
0.5
2021] will not be fair, because they be- long to a different type of mesh adap-
tation methods, and the downstream PDE solver will receive discretized problems with different topology and scale. Therefore, we compare our proposed models with two baselines that can be interpreted as ablated ver- sions of proposed models. The model MLP-Deform--ee Clip replaces the neural-spline block with MLP. The model GAT-Deform- Clip replaces the GAT-deformer with the ordinary GAT block. Instead of learning the positions of mesh nodes, we set the learning target as the mesh nodes displacement for the baseline models, because it gives better performance according to experimental results. In order to enforce that baseline models can also preserve boundary consistency, the nodes moved out of the boundary will be pulled back into the domain, and the displacement component perpendicular to the boundary is clipped for the nodes which are supposed to stay on the boundary, shown in Figure 3 However, there is straightforward way alleviate mesh tangling.
(a) Before clipping
(b) After clipping
Figure 3: Illustration of the mesh movement results of the baseline models before and after clipping.
4.1 Poisson's 's Equation
Poisson's 's equation second-order, linear, stationary PDE, which widely used in electrostatics and thermodynamies, amongst other fields. We consider solving class of 2-D Poisson's 's equations with different Dirichlet boundary conditions and mesh resolutions:
- Vu(z, v) f(x, v),
(z. ?
(6)
u(x; y) = n0( y), (x. E an.
For both the square and heptagonal domain experiments, we generate analytical samples from a mixed Gaussian distribution, which are fed into Poisson's 's equation to obtain the corresponding source terms f and boundary conditions up the problem samples, whereas the u functions serve as the ground truth.
Square Domain In this experiment, we train the models on cases with mesh resolution of 15 x 15 and 20 x 20, each with 275 samples. To evaluate the models and how well they generalize to different mesh resolutions, samples. Moreover, we deliberately set the optimal mesh movement to be drastic, in order to test how well different models can handle mesh tangling.
test cases with mesh resolution from 12 x 12 to 23 x 23, each with 125
The quantitative results are summarized in Table The proposed models, M2N-Spline and M2N- GAT, can achieve similar error reduction compared to the traditional MA method, while the mesh generation speed two to three orders of magnitude faster. addition, although the proposed models perform only marginally better than the two baseline models in enror reduction and time, they are proven very effective to keep the mesh untangled. In comparison, the baseline models suffer from mesh tangling.

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(b) M2N-Spline (c) HL---effrm-
(b) M2N- 12NiSpline (c) ALP-Deform
(34.3%)
Clip (illegal)
(14.98%)
Clip (10.40%)
(a)
MA
(a)
MA
(37.55%)
(33.35%)
(d) (32.05%)
(e) GAT-Deform
(d) M2N-GAT (e) GAT AAT-Deform- (28.33%)
Clip (illegal)
Clip (27.16%)
Figure 4: Comparison of mesh movement for an Figure 5: Comparison of mesh movement for an example problem of the Poisson's 's equation on example problem the Poisson's 's equation on the square domain. The percentage values in the heptagonal domain. The percentage values in the parentheses are the error reduction ratios.
parentheses are the error reduction ratios.
Table 2: Performance summary of the Poisson's 's equation problem the heptagonal domain.
Method
Error Reduction (%)
Time (ms)
Element Inversion (%)
MA (traditional)
26.57
7329.78
0.00
M2N-Spline
16.15 + 0.40
5.97 + 0.05
0,28
ML.P-Deform-Clip
16.49 + 0.46
2.60 + 0.02
0.56
M2N-GAT
22.39 + 0.27
9.41 + 0.03
0.00
GAT-Deform-Clip
22.40 + 0.47
10.81 + 0.07
0.68
An example is given in Figure 4 where the mesh density in the upper left corner needs to be very high (shown in Figure a)). is demonstrated that M2N-Spline is more flexible for the cases where the overall mesh deformation is required to be large. On the other hand, for M2N-GAT, because of the constrained movement in each layer to alleviate mesh tangling and the finite layer numbers, does not learn as well in such scenarios.
Irregular Heptagonal Domain To evaluate the performance of different models for more general domain shapes, we conduct an experiment using Poisson's equation in irregular heptagonal domain. The models are trained at mesh densities of 13, 16, 19, and 22, each with 320 samples, and tested on evaluate the performance and generalization
mesh densities from 12 to 23, each with 80 samples, capability of the models. The initial mesh 7 is generated with the Delaunay triangulation method provided by Gmsh [Geuzaine and Remacle 2009
The results are summarized in Table 2 It can be seen that all deep learning models can perform mesh movement around two three orders magnitude faster than the traditional MA method. Two GNN- based models perform better than the other two models, with the eтror reduction ratio comparable to the MA method. This is because the GNN-based models can naturally embed the information of the entire irregular domain into the network, and there are extra local feature extractors in the model, On the contrary, both the M2N-Spline and the MLP-Deform- -lip models are point-to-poing mappings with only the global feature extractor, hence lacking such capabilities. The results of all methods for an example are shown in Figure 5 from which we can see that the GNN-based models can better capture the delicate local structure where mesh resolution needs to be increased. Although mesh tangling still will not occur for M2N-GAT, it happens for M2N-Spline, because for the M2N-Spline model, its guarantee only works for hypercubic boundaries (rectangle in the 2-D case).

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Table 3: Performance summary of the Burgers' equation problem.
Method
Error Reduction (%)
Time (ms)
Element Inversion (%)
MA (traditional)
60.24
81590.64
0.00
M2N-Spline
48,92 + 1.33
5.54 + 0.02
0.00
ILLP-Defom- Clip
43.53 + 1.91
2.92 0.01
0.00
M2N-GAT
57.75 = 0.68
8.93 - 0.01
0.00
GAT-Deform-Clip
51.69 4.01
10.41 0.01
0.53
4.2 Burgers' Equation
The viscous Burgers' equation is non-linear, time-dependent PDE describing advection and diffusion processes in fluids. In this experiment, we consider class of 2-D Burgers' equations with different previous states, viscosity coefficients, and mesh resolutions:
+ (u Vyu- vV = 0,
(x. y) E Ω
Ot
(n V)u 0, (xr, y) E i2,
where constant scalar > 0 represents the viscosity coefficient and u is the velocity vector field obeying this PDE. We define the problem on the unit square domain Q [0,1]².
In this experiment, we still train the models on cases with mesh resolu- tion of 15 15 and 20 x 20, each with trajectories and timesteps per trajectory. and with different viscosity coefficients. Since a unit square domain is considered in this experiment, the initial uniform structured mesh Timi can be easily obtained by interpolation. To evalu- ate the models and how well they can generalize to different mesh resolutions, test cases with mesh resolution from 11 x 11 to 24 x 24, each with 8 trajectories and 60 timesteps per trajectory.
The results summarized in Table 3 It can be found that all learning- based methods are three to four ders of magnitude faster than the tra- ditional MA method. The accelera- tion is about one order magnitude larger than Poisson's is equation ex- periments, because the MA method slower for nonlinear PDE. Mesh tangling never occurs for M2N-Spline and M2N-GAT in this experiment. The GNN-based models perform better than the other two models, because they contain extra local feature extractors, and local information can be propagated better through edges, which im- portant for the Burgers' equation problem. Some generated mesh ex- amples at different timesteps of four
(a) MA
(b) M2N Spline
(c) M2N GAT
Figure 6: Comparison of mesh movement for the Burgers' equation problem. In each row a different sample.

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different trajectories are shown in Figure where it can be seen that M2N-GAT'S is better at performing delicate local deformation compared the M2N-Spline model.
Conclusion
In this paper, we have proposed the Mesh Movement Network (M2N), which to the best of our knowledge is the first learning-based end-to-end mesh movement method for PDE solvers. Traditional mesh movement methods can improve the accuracy of numerical PDE solutions without modifying the topology of the mesh, at the expense of solving auxiliary PDE, which is often very computationally expensive and sometimes makes the approach infeasible. With the power of deep learning, M2N generates adapted meshes for different PDE problems of the same type, with the solution precision comparable to ground truth but at a much faster speed. To achieve this robustly, we have designed a Neural-Spline based and GAT based mesh deformer, to guarantee the output adapted mesh retains boundary consistency, alleviates mesh tangling, and generalizes to different mesh densities. The results validated on the static linear Poisson's equation with regular and irregular domains, and the time-dependent nonlinear Burgers' equation.
On the other hand, there are still several limitations and future directions worth discussing. First of all, although the proposed Neural-Spline based and GAT based models can effectively alleviate the mesh tangling issue, they cannot theoretically guarantee that. Therefore, analyses of mesh tangling avoidance and error reduction improvement can be conducted. Secondly, in this paper, we used the method to generate the supervised optimized meshes, while there other traditional r-adaptation methods that can also be considered and tested. Finally, the scale and complexity of the current experiments are still far from practical applications. Experiments with more complicated boundary shapes, larger scales, in higher dimensions, and for more diverse PDE types, can be conducted to better validate the proposed methods.
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12


Reviews:
[{"review_id": "aUZtdJVlRzY", "reviewer": "Reviewer_2ifw", "summary": "The paper proposes learning-based mesh movement methods for improving the accuracy of PDE solutions. The key idea is to train neural networks to approximate the solutions generated using the traditional Monge-Ampère method. The models are evaluated using two types of PDE problems on different domains, where the proposed model achieved performance close to the traditional method while being much faster.\n\nThe main contribution of the paper is the proposed neural models that can quickly produce a mesh movement result close to the traditional Monge-Ampère method.", "questions": "1) What type of CPU were the experiments run on? Are the implementation of the MA method single-threaded or paralellized?\n\n2) What types of meshes are good for PDE solving? Why is not possible to encode these quality meshes into the loss function?\n\n3) How do the models perform on large meshes and non-planar domains?", "limitations": "The discussion is sufficient.", "rating": 4, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n- The proposed models can achieve performance close to the traditional Monge-Ampère method while being much faster.\n- The models are evaluated on two types of PDE problems to show their performance.\n\nWeakness:\n- According to the literature cited in the paper, the Monge-Ampère method has been applied to non-planar domains such as spheres. But the evaluation in this paper is only performed on planar domains. It is unclear how well the method performs on domains with non-zero Gaussian curvature.\n- Whilst the speed-up reported in the paper is impressive, they can also be misleading. The traditional MA method is run on the CPU, whilst the proposed models are run on the GPU. In the paper there is no information on the CPU model. Moreover, it is unclear what types of underlying linear algebra library the Firedrake code uses (NumPy can use different underlying BLAS libraries, which may have very different performance). So it is difficult to assess the actual speedup.\n- The proposed models are formulated purely based on closeness to the Monge-Ampère solution. However, Monge-Ampère is not the ultimate goal of the mesh movement problem, but rather just one particular way to generate high-quality meshes. The paper does not provide much information on what types of meshes are good for PDE solving, which makes it difficult to judge the quality of the result beyond its similarity to the Monge-Ampère solution. This somewhat limits the novelty of the proposed approaches.\n- Since the loss function is purely based on closeness to the Monge-Ampère solution rather than specific quality measures of the meshes, it cannot totally prevent mesh tangling, as mentioned in the paper. \n- The evaluation is done on small meshes. It is unclear how well the method performs on larger meshes (say more than 10K vertices).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "VLaClXnZcgS", "reviewer": "Reviewer_AXGX", "summary": "The authors have presented a novel approach for r-adaptivity based adaption for PDEs using two different network architectures. The Neural spline-based model serves well to avoid mesh tangling while handling meshes with large deformation, and the GAT-based model can generalize for domains with arbitrary shapes. The authors use the Monge-Ampere method to train using supervised learning, and demonstrate the effectiveness of their method on stationary and time-dependent, linear and non-linear equations, as well as regularly and irregularly shaped domains.", "questions": "(1) In irregular domains, the sampling is done by using a bounding box and values outside the actual domain are set to 0. We have found that such an approach is problematic if it used for solutions that have non-smooth (shocks) features. It would be good to note this limitation.\n(2) Can the boundary nodes move with the neural spline model? Node movement along the boundary can be crucial for certain applications.\n(3) Regarding the last two lines right before Section (4): Two adjacent nodes can move within the convex hull of their neighbors in such a way that the resulting mesh is tangled. Do you mean something else (e.g., half the distance from the closest neighbor)?", "limitations": "While this work is a step in the right direction, the proposed methods must really be demonstrated to work for large scale 2D/3D problems if they are to be claimed to be competitive against existing methods.", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strength:\n(1) The proposed framework is truly the first learning-based approach for r-adaptivity of meshes.\n(2) The authors have considered two different approaches in Neural splines and GAT, and demonstrated their effectiveness along with their limitations.\n\nWeaknesses:\nThe scope of the work seems somewhat limited in the sense of:\n(1) the authors consider  only the Monge Ampere method for r-adaptivity. See \"Simulation-driven optimization of high-order meshes in ALE hydrodynamics\", \"rp-adaptation for compressible flows\", and \"Defining a stretching and alignment aware quality measure for linear and curved 2D meshes\", which have been used for r-adaptivity of meshes to PDEs.\n(2) the problems considered are too simple; element count in each direction is ~20 and the simulations are 2D only.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "RwS-GJEZ13-", "reviewer": "Reviewer_h6GN", "summary": "The paper presents two neural architectures for warping meshes to improve finite element modeling performance (reduce error). The models are trained in a supervised manner to match the maps produced by a Monge-Ampere (MA) mesh movement method. Error reduction is comparable to the classical MA method, but with a speed improvement of several orders of magnitude.", "questions": "The examples shown in the paper are on relatively coarse, high-quality meshes of domains with convex boundaries. How does the method perform on larger meshes, or when the initial mesh is less regular? Can it be applied to other types of PDE? ", "limitations": "1. The models are only tested on two PDEs on simple domains. It would be good to see at least a few more evaluations, especially on larger/finer-grained meshes.\n2. Even though comparison to h-adaptation methods may not be useful to evaluate the performance of this method, it might still be useful for practitioners to choose which type of adaptation to use.", "rating": 7, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "The overall idea of using a normalizing flow to move a mesh around seems sound, and allows for a nice tradeoff between the speed advantages of neural methods and the reliability of traditional FEM. The neural spline model can only natively preserve boundaries that are aligned to the axes. Also, while the neural spline model may produce a diffeomorphism, this does not guarantee avoiding mesh tangling because the discrete mesh might not be able to represent a sufficiently complicated smooth diffeomorphism. For the graph attention model, the use of neighborhood attention to keep vertices within the convex hulls of their neighbors is a nice observation, and could potentially guarantee injectivity under suitable conditions (as with Tutte embedding).", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "7B716t7QtXy", "reviewer": "Reviewer_9Nru", "summary": "This paper proposes a new technique for moving the vertices of a mesh that is used as the underlying discretization for solving a PDE in order to better capture relevant details without increasing complexity. This is achieved by designing a neural spline-based, and also a graph attention network-based model to output the optimized node positions. Direct supervision coming from the Monge-Ampère method is utilized.", "questions": "- As far as I understand, with the GAT-based variant, it is not possible to guarantee non-self-intersecting meshes/mesh tangling. I was expecting an analysis of the behaviour of the generated maps for this characteristic, but there is none.\n\n- I was a bit disappointed by the irregular domain experiments, which is the main motivation for using GAT. Heptagon is not that irregular. I suggest including more examples with ideally real-world-like domains.", "limitations": "Please see above.", "rating": 6, "confidence": 2, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strenghths:\n\n- To my knowledge, this is the first method that proposes to optimize for node positions with direct supervision in the context of solving PDEs.\n- Can potentially inspire a line of research for re-meshing.\n\nWeaknesses:\n\n- Although the general technique is well-motivated, it is not clear why the particular choices of network architectures make sense. Most of the methods section is about listing the various design choices without justifications.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept