M2N: Mesh Movement Networks for PDE Solvers
Anonymous Author(s)
AffiliationAddress email
Abstract
Numerical Partial Differential Equation (PDE) solvers often require discretizingthe physical domain by using a mesh.
Mesh movement methods provide thecapability to improve the accuracy of the numerical solution without introducingextra computational burden to the PDE solver, by increasing mesh resolution wherethe solution is not well-resolved, whilst reducing unnecessary resolution elsewhere.
However, sophisticated mesh movement methods, such as the Monge-Ampèremethod, generally require the solution of auxiliary equations.
These solutions canbe 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-basedend-to-end mesh movement framework for PDE solvers.
Key requirements oflearning-based mesh movement methods are: alleviating mesh tangling, boundaryconsistency, and generalization to mesh with different resolutions.
To achievethese goals, we introduce the neural spline model and the graph attention network(GAT) into our models respectively.
While the Neural-Spline based model providesmore flexibility for large mesh deformation, the GAT based model can handledomains with more complicated shapes and is better at performing delicate localdeformation.
We validate our methods on stationary and time-dependent, linearand non-linear equations, as well as regularly and irregularly shaped domains.
Compared to the traditional Monge-Ampère method, our approach can greatlyaccelerate the mesh adaptation process by three to four orders of magnitude, whilstachieving comparable numerical error reduction.
Introduction
Partial Differential Equations (PDEs) are widely used to model natural phenomena, ranging fromastrophysics and ocean dynamics to semiconductor device simulation and bio-engineering [Evans,2010].
Acquiring accurate numerical solutions efficiently for complex PDEs is an essential butchallenging 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 itsquality affects the accuracy of the numerical solution [Frey and George, 2007].
However, it is oftenprohibitively expensive to solve the problem on a very high resolution mesh.
Mesh adaptation is anadvanced discretization method designed to tackle this problem.
It increases the mesh resolutionwhere the solution requires higher numerical accuracy, while decreasing the mesh resolution whereunnecessary.
Mesh adaptation methods can be generally divided into two categories: h -adaptationand r -adaptation.
In h -adaptation, new mesh nodes are dynamically added to the regions where fineresolution is required.
In r -adaptation (or mesh movement ), however, mesh nodes are only relocatedor 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 keepsthe 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 enablesmatrix pre-factorization to accelerate the solution of the large linear systems encountered in FEM[Budd et al., 2009].
However, a common problem of mesh movement methods is mesh tangling, inwhich lines connecting the mesh nodes come across each other.
Mesh movement methods based onoptimal transport theory can effectively prevent mesh tangling issues (see [Clare et al., 2022] for adiscussion on this), but require solving a Monge-Ampère equation at each adaptation step, which ishighly time-inefficient.
AI-powered approaches to computing solutions of PDEs have been an 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, Sheng and Yang, 2021]), or in accelerating the solutionprocess by learning a neural operator from the parameterized description of a PDE problem to itscorresponding solution [Li et al., 2020, Lu et al., 2019].
However, these methods still encounter fatalbottlenecks, such as the precision guarantee of the solution, data efficiency, generalization capability,etc.
These are fundamental limits of deep learning, but essential for scientific computing scenarios.
A 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 isgreatly 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., 2021, Pfaff et al., 2020].
However,most previous methods focus on mesh generation [Zhang et al., 2020] or mesh refinement [Yanget al., 2021, Huang et al., 2021, Fidkowski and Chen, 2021], instead of topology-invariant meshmovement 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 as the local mesh density[Zhang et al., 2020, Huang et al., 2021] or the metric tensor [Fidkowski and Chen, 2021, Tingfanet al., 2021, Pfaff et al., 2020], which then have to be fed into a traditional mesher/remesher toobtain the mesh.
Therefore, the overall performance is bounded by the mesher/remesher, which iscomputational 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 meshmovement framework for PDE solvers.
Taking the source term, the input field, and/or the PDEparameters as input features, the model deforms an initial mesh to the adapted mesh by meshmovement.
In usage, the model can be applied 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 canavoid mesh tangling.
Moreover, its mechanism naturally guarantees that a hypercubic boundary canbe maintained through learnable mapping.
We also design a graph attention network (GAT) basedmodel for mesh deformation.
The graph neural network can naturally describe domains with irregularshapes and embed the relevant information.
We utilize the attention mechanism of the GAT model toguarantee each mesh node stays within its neighborhood so that mesh tangling can be alleviated.
Our main contributions are listed as follows:
We propose a learning-based end-to-end mesh movement framework for PDE solvers, whichto 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-Ampèremethod, while the mesh generation is accelerated by three to four orders of magnitude.
A Neural-Spline based model and a 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, andgeneralize to different mesh densities, which are all desired for mesh movement.
Related Work
Mesh movement method.
Mesh movement methods include velocity-based and location-basedmethods.
In the work of [Anderson and Rai, 1983, Gnoffo, 1982, Farhat et al., 1998], the mesh ismoved according to attraction and repulsion pseudo-forces between nodes motivated by a springmodel.
The moving mesh finite element method [Baines et al., 2005] computes the solution and themesh simultaneously by minimizing the residual of the PDEs written in a finite element form.
As forlocation-based method, the moving mesh PDE (MMPDE) method [Huang and Russell, 2011] moves the mesh through the gradient flow equation of an adaptation functional.
In recent years, there hasbeen a growing interest in optimally-transported r -adapted meshes [Budd and Williams., 2009, Clareet al., 2022].
AI for PDE.
To solve a PDE problem, neural networks can be used to represent the function tosolve, and trained either with the residual loss of the PDE or using the variational principle [Raissiet al., 2019, Yu et al., 2017].
Neural operators are proposed to learn an operator from the problemfunction to the solution function [Li et al., 2020, Lu et al., 2019].
There also exist mesh-based PDEsolvers with deep learning.
In [Pfaff et al., 2020], a graph neural network with additional worldedges is applied to predict dynamical systems, shown to be effective with a wide range of physicalsystems.
In [Belbute-Peres et al., 2020], a differentiable PDE solver is embedded in a neural networkto help predict accurate solutions and also backpropagate the loss so that the input coarse mesh canbe optimized.
AI for Meshing.
AI methods have also been proposed for mesh generation, adaptation, and soon.
MeshingNet [Zhang et al., 2020] uses a neural network to learn the required local mesh density,which can then be provided to a Delaunay triangulation based mesh generator to generate high-qualitymeshes.
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 a reinforcement learning problem in[Yang et al., 2021] to minimize the PDE solution error under given refinement budgets.
The flowfield is predicted by machine learning models to calculate the metric tensor so that the mesh can beoptimized accordingly [Tingfan et al., 2021].
The authors in [Fidkowski and Chen, 2021] focused onoptimal anisotropic meshes by predicting the desired element aspect ratio.
In [Pfaff et al., 2020], thesizing 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 dimensions.
Each specific sample P n ∈ P ( n = 1 , 2 , 3 , . . . )
is determined by PDE related information,such as source terms, boundary conditions, PDE parameters, solution fields, etc.
We use X n to parameterize such information, which will be fed into the neural network as raw features, so that the trainedmodel can generalize to a class of PDEs.
The sample-specific information X n = { p n , M n } can befurther categorized into global parameters p n and position-dependent parameters M n .
Dependingon the type of the PDE problem and the terms we expect the trained model to be able to generalize to,some examples of p n are wave number of the Helmholtz equation and viscosity coefficient of the
Burgers’ equation, and options of M n are source terms, system states in the previous timestep fortime-dependent equations, etc.
Meshing is the procedure to spatially discretize a domain, which is necessary for most numericalmethods to solve PDEs.
Among these methods, the Finite Element Method (FEM) has been widelyused in various engineering fields, which we will take as our solving method in the experiments.
Theinitial mesh is generated with a traditional mesh generator.
We denote the initial mesh by T init and themesh node positions in T init by ξ init .
Since the generation of the initial mesh is purely computationalgeometry based, the sample-specific information X n cannot be utilized, hence T init is shared by allsamples P n ∈ P .
A standard PDE solver S will take as input the PDE problem P n discretized on themesh T init and output the corresponding solution u init n = S
( P n ( T init )) .
A high-quality adapted mesh can significantly improve the accuracy-efficiency trade-off of the PDEnumerical solution.
There are in general two types of mesh adaptation techniques.
In this work, weconsider mesh movement, or r -adaptation, which means we will not modify the number and topologyof mesh nodes, but only relocate them.
This characteristic provides the advantage that no remeshingneeds to be performed, and there will be less repetitive computation for the PDE solver since theinput matrices will maintain a constant size and sparsity structure
[Budd et al., 2009].
A typicalmesh movement method F will map the nodes ξ init in the initial mesh T init to ξ adap n = F ( ξ init , X n ) ,so that the adapted mesh T adap n will be constructed with ξ adap n and the topology defined on the initialmesh.
The PDE solution corresponding to the adapted mesh discretization u adap n = S ( P n ( T adap n ))is expected to be much more accurate than u init n , although the scale of the discretized problem that
PDE Solver 𝒮𝒮
Initial Mesh 𝒯𝒯 init
PDE Problem 𝒫𝒫 𝑛𝑛
Adapted Mesh 𝒯𝒯 𝑛𝑛M2N
Solution 𝒖𝒖 𝑛𝑛M2N
M2N Framework
GNN Block
GATDeformer
Global Feature Extractor
M2N - GAT
Graph Constructor (b) Mesh Movement Network Implementation (a) Framework Overview
Sample
Conv
GAP (c) Global Feature Extractor
NeuralSpline Global Feature Extractor
M2N - Spline
Global Parameters 𝒑𝒑 𝑛𝑛 𝝃𝝃 init
Global Parameters 𝒑𝒑 𝑛𝑛 𝒳𝒳 𝑛𝑛 𝒫𝒫 𝑛𝑛 ℳ 𝑛𝑛 𝑴𝑴 𝑛𝑛 𝑬𝑬 𝑛𝑛
MeshMovement Network ℱ M2N ℳ 𝑛𝑛 𝑬𝑬 𝑛𝑛 𝝃𝝃 init 𝝃𝝃 Spline 𝝃𝝃 GAT 𝒯𝒯 𝑛𝑛Spline 𝒯𝒯 𝑛𝑛GAT 𝝃𝝃 init ℳ 𝑛𝑛 𝑬𝑬 𝑛𝑛 𝑯𝑯 𝑛𝑛 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 ξ MA n = F MA ( ξ init , X n )will be taken as the supervised signal to train the neural network model, while the computationalefficiency and accuracy of the corresponding solution u MA n = S ( P n ( T MA n )) 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 theunderlying PDE problem, which is generally unacceptable.
Therefore, our goal is to model a Mesh
Movement Network (M2N) based mapping F M2N ( ·| θ ) , where θ represents the trainable parameters,such that the mesh adaptation process can be greatly accelerated.
The proposed framework isdemonstrated in Figure 1(a).
We consider constructing a learning-based mesh movement method thatrelocates the nodes ξ init in the initial mesh given the input state X n :
[Equation]
from which the adapted mesh T M2N n can be reconstructed based on the topology of the initial mesh.
From our empirical study, we take ℓ 1 loss between the model output and the adapted mesh nodesobtained with the Monge-Ampère method:
[Equation]
Under the M2N framework, we implemented two network structures: a Neural-Spline based networkand a GAT based network, which are denoted as M2N-Spline and M2N-GAT, respectively.
Bothmodels are designed to be capable of alleviating mesh tangling, keeping boundary consistency, andgeneralizing 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 globalmesh deformation is required, while M2N-GAT is able to handle irregularly shaped domains andcan better learn delicate local deformation.
The detailed model structures will be introduced in thefollowing sections, and more specifics of the model implementations can be found in Appendix A.
3.3 Neural-Spline based Network
As demonstrated in Figure 1(b), our Neural-Spline based network is mainly composed of two parts,where the input information X n = { p n , M n } will be separately fed into the model.
The globalfeature extractor GFE ( · ) extracts features from the position-dependent parameters M n to obtain the mesh resolution invariant embedding E n .
E n together with the global physical parameters p n willthen be fed into the neural spline deformer to control the mesh node relocation, i.e.,
[Equation]
where operator ⊕ represents tensor concatenation.
Global Feature Extractor As shown in Figure 1(c), the global feature extractor has three modules:
[Equation]
In detail, we uniformly sample the input state M n in the domain Ω .
If the domain is with an irregularboundary, sampling is performed inside its minimum bounding box and the values of the samplingpoints outside the domain boundary are set to zero.
The sampled states are assembled as a state tensor
M n .
To keep the extracted feature invariant to the magnitude, the state tensor M n is normalizedby its maximum absolute value.
After normalization, M n is sent into the convolutional layers forfeature extraction, whose output is further fed into a Global Average Pooling (GAP) layer
[Lin et al.
,2013] to obtain a mesh resolution invariant global embedding E n .
The embedding E n will then beconcatenated with global physical parameters p n to obtain I n , which is fed into the deformer.
Neural-Spline based Deformer Normalizing flow models [Kobyzev et al., 2020] are proposedto learn invertible mappings.
Neural spline
[Durkan et al., 2019], as a specific type of normalizing flows, transforms the input with a differentiable monotone rational-quadratic spline function
RQS ( ·| K ) , where K represents the learnable anchor points that determines the invertible mapping.
In our model, the neural spline deformer Spline ( ξ init , I n ) is a stack of neural spline layers
RQS d ( ξ ( d ) | K d ( I n , ξ ( − d ) )) , each of which transforms one dimension of the input node coordinatesξ ( d ) , whose anchor points K d are parameterized by the input features I n and the other dimensionsof node coordinates ξ ( − d ) .
Since the neural spline model is guaranteed to be an invertible mapping, mesh tangling can bealleviated.
A specialty of the neural spline model is that, the end points of the spline function arefixed, therefore the intervals of the input and the output can be kept unchanged.
In our case, thisproperty is utilized to preserve the mesh with a hypercubic boundary (e.g. a rectangle in the 2-Dcase).
Moreover, since the neural spline learns a continuous mapping, it can naturally generalize todifferent mesh resolution input.
3.4 GAT based Network
Although the Neural-Spline based network is well-designed for meshes of hypercubic domains, it isdifficult to extend to domains with more general boundaries.
Therefore, we also propose a Graph
Neural Network (GNN) based model, which can naturally handle irregular domains.
As shown in
Figure 1(b), the model consists of a two-branch feature extractor and a Graph Attention Network (GAT,[Veli ˇ ckovi ´ c et al., 2017]) based mesh deformer GAT ( · ) .
In this subsection, we omit the sample indexn when there is no misunderstanding, hence: F GATM2N ( ξ init , X ) = GAT ( ξ init , LFE ( M ) , GFE ( M ) ⊕ p ) .
Feature Extractor
The feature extractor of the GAT-based network consists of two branches.
Oneis the global feature extractor same as Eq.
(4), the output of which will also be concatenated withglobal 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 init .
For each graphnode index i ∈ { 1 , . . .
, | V |} , the input feature v (0) i is the sampling of the input state M at ξ init i , theposition of node i in the initial mesh T init .
To preserve the mesh density information, the node distance∥ ξ init i − ξ init j ∥ are encoded into edge features e (0) ij as the relative edge features.
The constructed graphwill then be processed by the GNN block with a message passing mechanism to propagate the localphysical information across the graph.
The edge features are updated by a Multi-Layer Perceptron(MLP)
f k ( · ) : e ( k ) ij ← f k ( e ( k − 1) ij , v ( k − 1)
i , v ( k − 1) j ) , and the node features are updated by summing upthe surrounding edge features v ( k ) i ← (cid:80) j ∈N ( i ) e ( k ) ij , where k ∈ { 1 , 2 , . . .
K GNN } means the layer index, and N ( i ) refers to the neighbors of node i .
At the output of the final layer, the features of eachgraph node are concatenated with the extracted global feature I to assemble H i = v ( K GNN )
i ⊕ I .
Weuse H
=
[ H 1 , . . .
H | V | ] to represent the extracted features of the entire graph.
[Figure]
GAT-based Deformer As shown in
Figure 2, the deformer consists of
K GAT GAT blocks.
The k -th blocktakes two inputs, mesh node positionsξ ( k − 1) and extracted features H ( k − 1) ,where H ( k − 1)
=
ˆ H ( k − 1) ⊕ ξ ( k − 1) .
For the first block, ˆ H (0) =
H andξ (0) =
ξ init .
The GAT block performsthe following transforms:
[Equation]
where α ( k ) ij is the attention score indicating the importance of node j to node i in the k -th block,acquired by the method introduced in [Veli ˇ ckovi ´ c et al., 2017], and W ( k ) is a learnable weightmatrix.
To keep the mesh shape consistent before and after deformation, the nodes originally on theboundary are restricted to move along the boundary of the domain ∂ Ω .
The adapted mesh T GAT willbe constructed from the output mesh node positions of the last block ξ ( K GAT ) .
Because of the attentionmechanism, during the mesh vertex relocation process, the movement of each mesh node is confinedinside the convex hull composed of its 1-ring neighbors, hence effectively alleviating mesh tangling.
Experiment
We evaluate our proposed models against two learning-based baseline models and the traditional
Monge-Ampère (MA) method, using error reduction ratio compared to the numerical solution onthe initial mesh, mesh adaptation time consumption, and element inversion ratio, to validate theirperformance, generalization capability, and robustness.
The experiments are conducted on thestationary and linear Poisson’s equation in both a square domain and an irregular heptagonal domain,and the time-dependent non-linear Burgers’ equation, where the supervised optimized meshes aregenerated with the traditional MA method.
Each experiment is run three times with different randomseeds to ensure the reliability of the model performances and provide mean and standard deviationof the results, which are summarized in the tables.
More details of the experimental setup, datasetgeneration, model training, and experimental results can be found in the Appendix. [Figure] [Figure] [Figure]
clipping
[Figure]
As mentioned in Section 1, there isno similar previous work to compareresults against.
Comparisons withlearning-based h -adaptation
[Pfaffet al., 2020, Zhang et al., 2020, Fidkowski and Chen, 2021, Huang et al.
,2021] will not be fair, because they belong to a different type of mesh adaptation methods, and the downstream
PDE solver will receive discretizedproblems with different topology andscale.
Therefore, we compare ourproposed models with two baselinesthat can be interpreted as ablated versions of our proposed models.
Themodel MLP-Deform-Clip replaces theneural-spline block with MLP.
Themodel GAT-Deform-Clip replaces the GAT-deformer with the ordinary GAT block.
Instead oflearning the positions of mesh nodes, we set the learning target as the mesh nodes displacement for
[Table]
[Figure]
[Figure]
the baseline models, because it gives better performance according to experimental results.
In orderto enforce that baseline models can also preserve boundary consistency, the nodes moved out of theboundary will be pulled back into the domain, and the displacement component perpendicular to theboundary is clipped for the nodes which are supposed to stay on the boundary, as shown in Figure 3.
However, there is no straightforward way to alleviate mesh tangling.
4.1 Poisson’s Equation
Poisson’s equation is a second-order, linear, stationary PDE, which is widely used in electrostaticsand thermodynamics, amongst other fields.
We consider solving a class of 2-D Poisson’s equationswith different Dirichlet boundary conditions and mesh resolutions:
[Equation]
For both the square and heptagonal domain experiments, we generate analytical u samples froma mixed Gaussian distribution, which are fed into Poisson’s equation to obtain the correspondingsource terms f and boundary conditions u 0 as the problem samples, whereas the u functions serve asthe ground truth.
Square Domain In this experiment, we train the models on cases with mesh resolution of 15 × 15and 20 × 20 , each with 275 samples.
To evaluate the models and how well they generalize to differentmesh resolutions, we test on cases with mesh resolution from 12 × 12 to 23 × 23 , each with 125samples.
Moreover, we deliberately set the optimal mesh movement to be drastic, in order to test howwell different models can handle mesh tangling.
The quantitative results are summarized in Table 1.
The proposed models, M2N-Spline and M2NGAT, can achieve similar error reduction compared to the traditional MA method, while the mesh
[Table] [Table]
generation speed is two to three orders of magnitude faster.
In addition, although the proposed modelsperform only marginally better than the two baseline models in error reduction and time, they areproven very effective to keep the mesh untangled.
In comparison, the baseline models suffer frommesh tangling.
An example is given in Figure 4, where the mesh density in the upper left corner needs to be veryhigh (shown in Figure 4(a)).
It is demonstrated that M2N-Spline is more flexible for the cases wherethe overall mesh deformation is required to be large.
On the other hand, for M2N-GAT, because ofthe constrained movement in each layer to alleviate mesh tangling and the finite layer numbers, itdoes not learn as well in such scenarios.
Irregular Heptagonal Domain To evaluate the performance of different models for more generaldomain shapes, we conduct an experiment using Poisson’s equation in an irregular heptagonal domain.
The models are trained at mesh densities of 13, 16, 19, and 22, each with 320 samples, and tested onmesh densities from 12 to 23, each with 80 samples, to evaluate the performance and generalizationcapability of the models.
The initial mesh T init is generated with the Delaunay triangulation methodprovided by Gmsh [Geuzaine and Remacle, 2009].
The results are summarized in Table 2.
It can be seen that all deep learning models can perform meshmovement around two to three orders of magnitude faster than the traditional MA method.
Two GNNbased models perform better than the other two models, with the error reduction ratio comparable tothe MA method.
This is because the GNN-based models can naturally embed the information of theentire irregular domain into the network, and there are extra local feature extractors in the model.
Onthe contrary, both the M2N-Spline and the MLP-Deform-Clip models are point-to-point mappingswith only the global feature extractor, hence lacking such capabilities.
The results of all methods foran example are shown in Figure 5, from which we can see that the GNN-based models can bettercapture the delicate local structure where mesh resolution needs to be increased.
Although meshtangling still will not occur for M2N-GAT, it happens for M2N-Spline, because for the M2N-Splinemodel, its guarantee only works for hypercubic boundaries (rectangle in the 2-D case).
4.2 Burgers’ Equation
The viscous Burgers’ equation is a non-linear, time-dependent PDE describing advection and diffusionprocesses in fluids.
In this experiment, we consider a class of 2-D Burgers’ equations with differentprevious states, viscosity coefficients, and mesh resolutions:
[Equation]
where constant scalar ν > 0 represents the viscosity coefficient and u is the velocity vector fieldobeying this PDE.
We define the problem on the unit square domain Ω = [0 , 1] 2 .
[Figure]
In this experiment, we still train themodels on cases with mesh resolution of 15 × 15 and 20 × 20 , eachwith 9 trajectories and 60 timestepsper trajectory, and with differentviscosity coefficients.
Since a unitsquare domain is considered inthis experiment, the initial uniformstructured mesh T init can be easilyobtained by interpolation.
To evaluate the models and how well theycan generalize to different meshresolutions, we test on cases withmesh resolution from 11 × 11 to24 × 24 , each with 8 trajectories and60 timesteps per trajectory.
The results are summarized in Table3.
It can be found that all learningbased methods are three to four orders of magnitude faster than the traditional MA method.
The acceleration is about one order of magnitudelarger than Poisson’s equation experiments, because the MA methodruns even slower for a nonlinear PDE.
Mesh tangling never occursfor M2N-Spline and M2N-GAT
inthis experiment.
The GNN-basedmodels perform better than the othertwo models, because they containextra local feature extractors, andlocal information can be propagatedbetter through edges, which is important for the Burgers’ equationproblem.
Some generated mesh examples at different timesteps of fourdifferent trajectories are shown in Figure 6, where it can be seen that M2N-GAT is better at performingdelicate local deformation compared to the M2N-Spline model.
Conclusion
In this paper, we have proposed the Mesh Movement Network (M2N), which to the best of ourknowledge is the first learning-based end-to-end mesh movement method for PDE solvers.
Traditionalmesh movement methods can improve the accuracy of numerical PDE solutions without modifying thetopology of the mesh, at the expense of solving an auxiliary PDE, which is often very computationallyexpensive and sometimes makes the approach infeasible.
With the power of deep learning, M2Ngenerates adapted meshes for different PDE problems of the same type, with the solution precisioncomparable to ground truth but at a much faster speed.
To achieve this robustly, we have designed a
Neural-Spline based and a GAT based mesh deformer, to guarantee the output adapted mesh retainsboundary consistency, alleviates mesh tangling, and generalizes to different mesh densities.
Theresults are validated on the static linear Poisson’s equation with regular and irregular domains, andthe time-dependent nonlinear Burgers’ equation.
Important future directions include theoretical analysis on mesh tangling avoidance and error reduction improvement, evolution of the Neural-Spline based model to better handle irregular domains,investigation into higher dimensional problems, etc.
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information or offensive content?
[N/A]
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount
spent on participant compensation?
[N/A]