Paper: Better Neural PDE Solvers Through Data-Free Mesh Movers (ICLR-2024)

Reviewer comment (Experiments):
One essential challenge for r-adaptive (i.e., mesh movement) based mesh adaptive methods is mesh tangling. The mesh tangling happens when the mesh have elements with negative Jacobian-determinant, or in other words, the edges of mesh nodes come across each other after being moved. Although the optimal-transport based Monge-Amp`ere mesh movement method is designed to alleviate the mesh tangling issue, it is hard to guarantee tangling-free especially for a learned model, which only has a soft physical loss as the constraint. It is unclear how the proposed DMM perform regarding mesh tangling as there are no related evaluations in the paper.

Author response:
**Answer**: Thanks for your suggestion. 

- Firstly, as we use the output of DMM to enhance neural solvers' message passing rather than use it as generated mesh for traditional PDE solvers, mesh tangling does not pose a problem in our context. 
- Secondly, to address your concern, we also made an additional discussion and exploration on it in Section 6.4 and Appendix C.2.  Constraining the Jacobian determinant of $f$ to be non-negative is equivalent to requiring $\Phi$ in our paper to be convex, which is already constrained by $l_{convex}$. The trend of $l_{convex}$ during training is illustrated in Figure 3 in our paper, showing that it can reach a very low level (nearly zero) early in the training process. Additionally, we find that the generated grid points of test data have $l_{convex}=0$. These demonstrate that this constraint is well met.

Reviewer comment (Experiments):
The Monge-Amp`ere based physical loss is the key contribution of this paper, however it is not clear how each part of the loss i.e. $loss_{equation}$, $loss_{bound}$ and $loss_{convex}$ contribute to the model performance.

Author response:
**Answer**: Thanks for your question. We conduct new experiments to explore how the three losses impact the performance of DMM in Section 6.4 and Appendix C.3. We reduce the weights of the three losses within the physics loss by a factor of ten for training respectively. Results show that when the weight of $l{equation}$ is decreased, the std and range increase the most, suggesting that the MA equation is not adequately satisfied. The result does not deviate significantly from the original when the weight of $l_{convex}$ is decreased, as $l_{convex}$ can already reach a low level as stated in Answer to Re 1. Figure 4 in the paper shows the plotted meshes of decreased $l_{bound}$, implying that boundary conditions are not as well satisfied as before. All the results above conform to intuition.

Reviewer comment (Experiments):
The experiments are limited to one fixed mesh resolution (48 x 48 or 2521 triangular lattices) and there are no evaluations on generalization of the proposed model, i.e., trained on a (small) resolution and test on other (potentially larger) resolution. Therefore, it is not clear how the proposed method perform given unseen data / boundary conditions/ physical parameters in equations.

Author response:
**Answer**: Thanks for your question. One of DMM's advantages is its ability to generate meshes of varying resolutions during inference. 

- We have added a new experiment in Section 6.4 and Appendix C.1. The result shows that DMM can handle data of resolution different from the training data's resolution.
- Please note that since the selected encoder 1 cannot handle varying resolutions, we need to interpolate the state $u$ prior to inputting states of different resolutions. We assess bicubic interpolation and the softmax interpolation as in Eq 16, with training data at $48\times48$ and 2521 resolutions. 
- The meshes from the pre-trained DMM are illustrated in Figure 2 in the paper, and associated quantitative results are in Table 7 and 8 in the paper. Results show that the data adheres well to the MA equation regardless of resolution.

Reviewer comment (Writing):
The presentation and text of the paper should be improved.

Author response:
**Answer:** Thanks for your suggestion. We have modified our paper according to the reviewers' suggestions and revised the entire paper.

Reviewer comment (Experiments):
The mesh tangling is a key challenge for mesh movement methods. It is hard to evaluate the performance of the proposed mesh mover without the evaluations on this potential issue. I would suggest that adding a solid discussion for mesh tangling will improve the soundness of this paper.

Author response:
**Answer**: Thanks for your suggestion. 

- Firstly, as we use the output of DMM to enhance neural solvers' message passing rather than use it as generated mesh for traditional PDE solvers, mesh tangling does not pose a problem in our context. 
- Secondly, to address your concern, we also made an additional discussion and exploration on it in Section 6.4 and Appendix C.2.  Constraining the Jacobian determinant of $f$ to be non-negative is equivalent to requiring $\Phi$ in our paper to be convex, which is already constrained by $l_{convex}$. The trend of $l_{convex}$ during training is illustrated in Figure 3 in our paper, showing that it can reach a very low level (nearly zero) early in the training process. Additionally, we find that the generated grid points of test data have $l_{convex}=0$. These demonstrate that this constraint is well met.

Reviewer comment (Experiments):
Regarding to the physical loss, the $loss_{equation}$ part is designed based on the difference between the current mesh and the uniform mesh. In a time-dependent problem, does it indicate the mesh mover always move the mesh nodes starting from the very initial uniform mesh without using the moved mesh in the last timestep? Does this strategy perform better comparing to reuse the previous moved mesh as the starting mesh?

Author response:
**Answer**: Thanks for your question. The reason why we can not learn the meshes based on meshes in the last timestep is that the meshs in the last timestep are not known explicitly at the DMM training period, which means that we can not deduce the physics loss in this situation.

Reviewer comment (Experiments):
An ablation study for contributions of each components in the physical loss ($loss_{equation}$, $loss_{bound}$ and $loss_{convex}$) will help evaluate its effectiveness.

Author response:
**Answer**: Thanks for your question. We conduct new experiments to explore how the three losses impact the performance of DMM in Section 6.4 and Appendix C.3. We reduce the weights of the three losses within the physics loss by a factor of ten for training respectively. Results show that when the weight of $l{equation}$ is decreased, the std and range increase the most, suggesting that the MA equation is not adequately satisfied. The result does not deviate significantly from the original when the weight of $l_{convex}$ is decreased, as $l_{convex}$ can already reach a low level as stated in Answer to Re 1. Figure 4 in the paper shows the plotted meshes of decreased $l_{bound}$, implying that boundary conditions are not as well satisfied as before. All the results above conform to intuition.

Reviewer comment (Experiments):
Mesh movement methods may struggle with cases where the monitor functions give large values near boundaries, e.g. the peak values of the initial conditions of the burger's equation are located close to simulation domain boundary. It would be more convincing if there are such test cases for evaluating the proposed model.

Author response:
**Answer**: Thanks for your question. A new visualization of these cases is provided in Appendix D. We choose some states where the maximum value of the monitor function is relatively close to the boundary to generate meshes. Additionally, we have generated new states with maximum values much closer to boundaries and obtained the new mesh. We can see from Figure 7 that DMM can also handle such situations.

Reviewer comment (Experiments):
It is mentioned in the paper that the mesh for Burger's is 48x48, however it is only 20x20 shown in Figure 2. Are there any missing results for the 48 x 48 resolution?

Author response:
**Answer**: Thanks for your question.  We have added 48x48 meshes to Appendix D.
The reason why we show 20x20 meshes previously is that 20x20 meshes are clearer and can help reduce the length of the main text.

Reviewer comment (Evaluation):
Runtimes should be mentioned for the proposed model, especially when comparing with equivalent methods such as GNN, CNN, FNO and LAMP. How much overhead does DMM require, and what number of extra trainable parameters are we talking about in practice?

Author response:
**Answer**: Thanks for your suggestions. We have added the inference time of MM-PDE to Table 1 in Section 6.1 and list it below.  We can see that MM-PDE is comparable to GNN. But the time is slower CNN and FNO since the base model architecture GNN is slower.

| Model   | MM-PDE (MP-PDE) | MP-PDE | bigger MP-PDE | MM-PDE (MGN) | MGN    | CNN    | FNO    | LAMP   |
| ------- | --------------- | ------ | ------------- | ------------ | ------ | ------ | ------ | ------ |
| Time(s) | 0.5192          | 0.3078 | 0.3298        | 0.1187       | 0.0400 | 0.0027 | 0.0159 | 1.4598 |



> **Question about the overhead DMM requires.**

**Answer**: Thank you for your questions, we have revised Section 6.1 and 6.2 to add the information to the paper for clarity.

- As for the overhead DMM requires, the DMM for the Burgers' equation has a parameter count of 1222738, with a training time of 7.52 hours and memory usage of 15766MB. The DMM for the flow around a cylinder has 2089938 parameters, with a training time of 22.91 hours and memory usage of 73088MB. And the inference of the DMM is 0.001s for the Burgers' equation and 0.088s for the flow around a cylinder, while several tens of seconds are needed for the conventional PDE solver on the Burgers' equation's mesh adaptation. [1] This means that once DMM is trained, the use of it is quite fast and the time exceeds traditional solvers a lot.  

[1] Song W, Zhang M, Wallwork J G, et al. M2N: mesh movement networks for PDE solvers[J]. Advances in Neural Information Processing Systems, 2022, 35: 7199-7210.

Reviewer comment (Evaluation):
When training the DMM separately, how well is the physics loss being minimized? It is important to know to what extent the equations need to be satisfied until we see an improvement in MM-PDE.

Author response:
**Answer**: Thank you for your question. 

- We have revised Section 6.4 and provided the physics loss of DMM at the beginning and end of the Burgers' equation and flow around a cylinder both below. It is evident that there is a substantial decrease in this loss throughout the training process, indicating that the DMM has been optimized effectively. The result is also listed below.
- The reason we did not previously provide this loss is that we perceived it as not being sufficiently intuitive. Therefore, we proposed the std and range as more intuitive indicators before.  

| Loss           | Init    | Adam    | BFGS    |
| -------------- | ------- | ------- | ------- |
| $l$            | 28.248  | 0.045   | 0.026   |
| $l_{equation}$ | 28.209  | 0.044   | 0.023   |
| $l_{equation}$ | 3.94e-4 | 1.07e-6 | 2.80e-6 |
| $l_{convex}$   | 2.47e-7 | 0.0     | 0.0     |

| Loss           | Init    | Adam    | BFGS    |
| -------------- | ------- | ------- | ------- |
| $l$            | 0.881   | 0.101   | 0.095   |
| $l_{equation}$ | 0.547   | 0.095   | 0.093   |
| $l_{bound}$    | 3.34e-4 | 5.99e-6 | 1.54e-6 |
| $l_{convex}$   | 0.0     | 0.0     | 0.0     |

Reviewer comment (Experiments):
A sentence of explanation should be added when training the DMM, where we use a combination of both Adam and BFGS, but only finetune the last layer. Did this show the best performance? Why did Adam alone not suffice. When the high memory consumption is mentioned, what numbers is required by the two example problems shown in this paper?

Author response:
**Answer**: Thanks for your question. 

- Our choice to combine the Adam and BFGS optimizers follows many works on Physics-Informed Neural Networks (PINNs) that use several rounds of the LBFGS optimizer at the end of the optimization process [1, 2, 3]. However, in practice, we find that the LBFGS optimizer is quite slow and not very accurate, failing to further reduce the loss. Consequently, we employ the more precise BFGS optimizer. 
- To address your concern, we also have conduced an additional ablation study to show whether the BFGS is effective. The result is also listed in Table 2 in Section 6.4 and the table above. We can conclude that BFGS indeed results in a further reduction of the DMM loss.
- However, the BFGS optimizer does require more memory, so we only optimize the parameters of the final layer, and we try to increase the width of this layer as much as possible. The current memory usage has already been mentioned earlier: the Burgers' equation requires 15766MB, and the flow around a cylinder requires 73088MB. A single GPU card would not be sufficient to support our training if we optimized the parameters of the last two layers. However, it can be anticipated that optimizing more parameters could potentially improve the optimization results.

[1] Lou Q, Meng X, Karniadakis G E. Physics-informed neural networks for solving forward and inverse flow problems via the Boltzmann-BGK formulation[J]. Journal of Computational Physics, 2021, 447: 110676.

[2] PATRONI R. Towards adaptive PINNs for PDEs: a numerical exploration[J]. 2022.

[3] Pratama D A, Abo-Alsabeh R R, Bakar M A, et al. Solving partial differential equations with hybridized physic-informed neural network and optimization approach: Incorporating genetic algorithms and L-BFGS for improved accuracy[J]. Alexandria Engineering Journal, 2023, 77: 205-226.

Reviewer comment (Experiments):
From a practitioner's perspective, it is unclear why the metric for cell volume matters for the examples, aside from showing that the DMM loss was minimized. Most importantly, we never perform any testing on the adapted/moved mesh, hence its structure can be arbitrary as long as it is conducive to the training of the GNN. It has been shown that uniform meshes as a representation perform worse, but is it the case that reducing the cell volume std and range will also monotonically decrease the test loss of the model?

Author response:
**Answer**: Thanks for your question. 

- Firstly, the optimal moving mesh is the one satisfying the equidistribution principle under optimal monitor function *M*, which means mesh movement uniformizes cells’ volumes under *M*. Thus we calculate the standard deviation (std) and range of all cells’ volumes as the evaluation metric of DMM.

- Secondly, to further show how the std and range of meshes influence the MM-PDE, we conduct a new ablation study in section 6.4. Since the std and range are hard to directly control, we let the weighted sum of identity map $f_I$ and the DMM's coordination transformation map $\tilde{f}$ replace $\tilde{f}$, i.e. $\alpha\tilde{f} + (1-\alpha)f_I$. This operation means the new meshes' movement is between the moving meshes' movement from DMM and uniform meshes. The results are shown in the table below, illustrating the monotonic relationship between std, range and MM-PDE's loss. 

| $\alpha$ | std    | range  | Error        |
| -------- | ------ | ------ | ------------ |
| 0.8      | 0.0581 | 0.3432 | **6.154e-7** |
| 0.6      | 0.0708 | 0.4685 | 8.714e-7     |
| 0.4      | 0.0826 | 0.5695 | 9.209e-7     |
| 0.3      | 0.0939 | 0.6636 | 1.224e-6     |

Reviewer comment (Experiments):
For the non-uniform grid example in flow past cylinder, CNN and FNO were not considered. Would the non-uniform equivalent of FNO, the Geometry-Aware FNO GeoFNO, be applicable as a baseline? If so, comparing with non-uniform state-of-the-art methods would strengthen the paper as well.

Author response:
**Answer**: Thanks for your suggestion. According to your requirements, we have added a new baseline Geo-FNO on the flow around a cylinder. New results of this experiment are in the table below. Our MM-PDE still performs the best. This has been added to Section 6.2.

| Model | MM-PDE     | GNN    | bigger GNN | CNN  | FNO  | LAMP   | Geo-FNO |
| ----- | ---------- | ------ | ---------- | ---- | ---- | ------ | ------- |
| Error | **0.0846** | 0.2892 | 0.1548     | -    | -    | 0.4040 | 0.2844  |

Reviewer comment (Evaluation):
More details should be added on the experimental results, for example, for the flow past cylinder, is the MSE computed on just the velocity, or also pressures? The values reported in Table 2 seem relatively high then, perhaps visualization of the rollout results could also be added to the appendix.

Author response:
**Answer**: Thanks for your question. 

- Firstly, the error is computed on the horizontal velocity. We have revised Section 6.2 to make it clear. 
- Secondly, we also compute the relative MSE as below. We can see that the RMSE is relatively low and MM-PDE still surpasses a lot.  This has been added to Appendix E.1.

| Model | MM-PDE | GNN    | bigger GNN | LAMP   | Geo-FNO |
| ----- | ------ | ------ | ---------- | ------ | ------- |
| Error | 0.0197 | 0.0649 | 0.349      | 0.0693 | 0.0602  |

- Thirdly, the visualization of the rollout results of this experiment has been added to Appendix E.1's Figure 8, showing that the data produced by MM-PDE aligns well with the real trajectory.

Reviewer comment (Experiments):
The implications of the ablation study are that having a trainable DMM performs better, but having it trainable also means now it is finetuning on the MSE loss from the solution data, hence the DMM equations may no longer hold. Did you quantify the std and range of cell volumes after this mm+end2end training as well?

Author response:
**Answer**: Thanks for your question. We find that after mm+end2end training, the std and range of cell volumes under the monitor function we calculated before are not lower. But we also obtain that the monitor function calculated now is not very accurate, since we only get the discrete value of $u$ on the grids and we take the finite difference method to compute the derivatives in the monitor function.

Reviewer comment (Reproducibility):
How are the DMM loss derivatives computed, using finite differences, or autodifferentiability?

Author response:
**Answer**: Thanks for your question. We only use the finite difference method to compute the monitor function, as we only obtain the discrete $u$. Then, the other derivatives in physics loss $l$ are all computes using autodifferentiability such as $\frac{\partial \Phi}{\partial x}$ and so on. This explanation has been added to the **Physics loss** part in Section 5.1.

Reviewer comment (Experiments):
Is there any limitation in terms of extending the framework to 3D problems? Computational complexity?

Author response:
**Answer**: Thanks for your constructive question. We have conducted a new experiment on 3d GS reaction-diffusion equation to show that there is no difficulty for our method to extend to higher dimensions. Results in the below table indicate that our proposed method still works in 3d cases. This has been added to Section 6.3.

| Model | MM-PDE       | GNN      |
| ----- | ------------ | -------- |
| MSE   | **1.852e-4** | 2.106e-4 |

Reviewer comment (Experiments):
The efficiency of DMM requires further comprehensive demonstration.

Author response:
> **Concerns about the efficiency of DMM.**

**Answer**: Thanks for your question. To make it clear, we have revised Section 6.1 to show that the Data-free Mesh Mover (DMM) is fast in inference. 

- The DMM is an operator trained on data belonging to a distribution rather than a single state. Therefore, once training is complete, it requires only a brief inference time when applied to new data from the same distribution.

- Regarding DMM: we have calculated the inference time of the DMM to be 0.001s for the Burgers' equation and 0.088s for the flow around a cylinder, whereas the conventional PDE solver for the Burgers' equation's mesh adaptation requires several tens of seconds [1].

- Regarding MM-PDE: when we use MM-PDE, the extra cost brought by DMM is comparable with the inference time. A comparison of the MMPDE's time with other baselines' is provided in the table below, showing that its time is comparable to other models. And we have added these above results to the new version of our paper in the section of the 'Burgers' equation'.

| Model   | MM-PDE (MP-PDE) | MP-PDE | bigger MO-PDE | MM-PDE (MGN) | MGN    | CNN    | FNO    | LAMP   |
| ------- | --------------- | ------ | ------------- | ------------ | ------ | ------ | ------ | ------ |
| Time(s) | 0.5192          | 0.3078 | 0.3298        | 0.1187       | 0.0400 | 0.0027 | 0.0159 | 1.4598 |

[1] Song W, Zhang M, Wallwork J G, et al. M2N: mesh movement networks for PDE solvers[J]. Advances in Neural Information Processing Systems, 2022, 35: 7199-7210.

Reviewer comment (Novelty):
The contribution of this work appears to be incremental, as the core neural PDE solver used is essentially the message passing neural PDE solver developed by Brandstetter et al.

Author response:
> **Concerns for the core contribution.**

**Answer**: Thanks for your discussion about the contribution. Please note that our contribution includes design on both DMM and MM-PDE rather than choosing a specific neural network structure for MM-PDE. 

- To further demonstrate that our contribution is orthogonal to the neural network structure in MM-PDE, we have conducted an additional experiment using another model architecture Mesh Graph Nets commonly used to handle irregular grids [1]. This is to showcase the universality and novelty of our method. The table below shows the experimental results of the Burgers' equation. From the table, the error of MGN combined with the MM-PDE framework still surpasses MGN, which indicates the effectiveness of our method. The results are added to the 'Burgers' equation' section.

| Model | MM-PDE (MGN) | MGN      |
| ----- | ------------ | -------- |
| Error | **2.138e-5** | 2.459e-5 |

- For clarity, we have revised our conclusion section to make our contribution clearer: As for DMM, we introduce the physics loss to make it data-free, and we design the model architecture, sampling strategy, and training framework to enhance performance. As for the MM-PDE, we propose a framework to improve the base model and have added many design features tailored specifically for the moving meshes, such as the two-branch architecture, the residual cut network, and interpolation networks capable of adaptively generating interpolation weights. Given the dynamic and irregular grid, we thus choose the GNN network structure employed by Brandstetter et al. as our base architecture. 

[1] Pfaff T, Fortunato M, Sanchez-Gonzalez A, et al. Learning mesh-based simulation with graph networks[J]. arXiv preprint arXiv:2010.03409, 2020.

Reviewer comment (Experiments):
My main concern is on the efficiency of DMM. As data-free, for each input solution \(u\), training is needed. Then this can be very costly. Can authors compare the time if using the conventional PDE solver for the mesh adaptation?

Author response:
> **Concerns about the efficiency of DMM.**

**Answer**: Thanks for your question. To make it clear, we have revised Section 6.1 to show that the Data-free Mesh Mover (DMM) is fast in inference. 

- The DMM is an operator trained on data belonging to a distribution rather than a single state. Therefore, once training is complete, it requires only a brief inference time when applied to new data from the same distribution.

- Regarding DMM: we have calculated the inference time of the DMM to be 0.001s for the Burgers' equation and 0.088s for the flow around a cylinder, whereas the conventional PDE solver for the Burgers' equation's mesh adaptation requires several tens of seconds [1].

- Regarding MM-PDE: when we use MM-PDE, the extra cost brought by DMM is comparable with the inference time. A comparison of the MMPDE's time with other baselines' is provided in the table below, showing that its time is comparable to other models. And we have added these above results to the new version of our paper in the section of the 'Burgers' equation'.

| Model   | MM-PDE (MP-PDE) | MP-PDE | bigger MO-PDE | MM-PDE (MGN) | MGN    | CNN    | FNO    | LAMP   |
| ------- | --------------- | ------ | ------------- | ------------ | ------ | ------ | ------ | ------ |
| Time(s) | 0.5192          | 0.3078 | 0.3298        | 0.1187       | 0.0400 | 0.0027 | 0.0159 | 1.4598 |

[1] Song W, Zhang M, Wallwork J G, et al. M2N: mesh movement networks for PDE solvers[J]. Advances in Neural Information Processing Systems, 2022, 35: 7199-7210.

Reviewer comment (Reproducibility):
For time-dependent problems, what is the input \(u\)? At the current time step or the next time step? The singularity of the solution could move. The mesh obtained for \(u_k\) may not be suitable for \(u_{k+1}\).

Author response:
> **Question about only using the current state for moving mesh.**

**Answer**: Thank you for this question. In practice, we use the state of the current time step as input. 

- On the one hand, PDE solvers can only see the state of the current time step during the solving process. 
- On the other hand, to the best of our knowledge, traditional methods that use r-adaptive meshes also generate adaptive meshes based on the current time step [1, 2, 3].  
- However, we also considered the issue you raised before. We may be able to give the mesh mover the ability to predict the future by introducing a surrogate model to provide information about the future. We had originally intended to explore this idea as a topic for future work. Now we have incorporated our thoughts on this issue into the 'Conclusion and Discussion' section of the manuscript.

[1] Budd C J, Cullen M J P, Walsh E J. Monge–Ampére based moving mesh methods for numerical weather prediction, with applications to the Eady problem[J]. Journal of Computational Physics, 2013, 236: 247-270.

[2] Wallwork J G, Barral N, Ham D A, et al. Goal-oriented error estimation and mesh adaptation for tracer transport modelling[J]. Computer-Aided Design, 2022, 145: 103187.

[3] Budd C J, Williams J F. Parabolic Monge–Ampère methods for blow-up problems in several spatial dimensions[J]. Journal of Physics A: Mathematical and General, 2006, 39(19): 5425.

Reviewer comment (Experiments):
The error listed in Table 1 shows MM-PDE only improves from around \(2 \times 10^{-6}\) to \(6.7 \times 10^{-7}\) and in Table 2 is from \(0.0258\) to \(0.0141\) using a bigger GNN. Then a fair comparison is the computational time of MM-PDE and the bigger GNN to achieve the same error. A bigger GNN may need a larger model size, but since no DMM solver is involved, the overall time might be smaller. Again, this is related to Question 1: the efficiency of DMM.

Author response:
> **Concerns about the efficiency of DMM.**

**Answer**: Thanks for your question. To make it clear, we have revised Section 6.1 to show that the Data-free Mesh Mover (DMM) is fast in inference. 

- The DMM is an operator trained on data belonging to a distribution rather than a single state. Therefore, once training is complete, it requires only a brief inference time when applied to new data from the same distribution.

- Regarding DMM: we have calculated the inference time of the DMM to be 0.001s for the Burgers' equation and 0.088s for the flow around a cylinder, whereas the conventional PDE solver for the Burgers' equation's mesh adaptation requires several tens of seconds [1].

- Regarding MM-PDE: when we use MM-PDE, the extra cost brought by DMM is comparable with the inference time. A comparison of the MMPDE's time with other baselines' is provided in the table below, showing that its time is comparable to other models. And we have added these above results to the new version of our paper in the section of the 'Burgers' equation'.

| Model   | MM-PDE (MP-PDE) | MP-PDE | bigger MO-PDE | MM-PDE (MGN) | MGN    | CNN    | FNO    | LAMP   |
| ------- | --------------- | ------ | ------------- | ------------ | ------ | ------ | ------ | ------ |
| Time(s) | 0.5192          | 0.3078 | 0.3298        | 0.1187       | 0.0400 | 0.0027 | 0.0159 | 1.4598 |

[1] Song W, Zhang M, Wallwork J G, et al. M2N: mesh movement networks for PDE solvers[J]. Advances in Neural Information Processing Systems, 2022, 35: 7199-7210.

Reviewer comment (Writing):
"Two-order" should be corrected to "second order."

Author response:
> **Question about the typos.**

**Answer**: Thank you for raising this issue. We have revised the entire paper to fix the typos and enhance its readability.