Venue: NeurIPS 2022 Conference

Reviews:

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: This paper proposes an algorithm to solve PDEs that's agnostic to the input mesh. Precisely, the proposed method follows the "encoder-interpolation-forcast" pipeline. This pipeline will first extract features for each of the sampled mesh points, then these features will be passed into the interpolator to create features for the query points. The query points features will passed through the forcaster to make the final prediction. The experimental results show that MAgNet is capable of performing un par with baselines such as FNO and MPNN. The results also show the proposed interpolation in feature space does lead to improvement. The results also show that the model is able to test on a mesh resolution that's different from the trained mesh resolution.
strengths_and_weaknesses: Strengths:
I think the main strength of the proposed idea is that they did interpolation and interaction in a learned feature space instead of in the physics space directory. This allows the encoder and the interpolator to work together to make a potentially better interpolation schema which is agnostic to the sampling of the mesh vertices. The ablation studies seem to suggest that the learned interpolation is indeed bringing some advantages (per figure 4(a)). I suspect such advantage will be larger (i.e. comparing to Cubic interpolation on the original frames) when it comes to more irregular input meshes. 

Weakness:
1. Is the model actually "mesh agnostic"? My understanding of "mesh-agnostic" is that if both meshes represent the same geometry, then the method should output the same results. The experiments seem to show that if two meshes are representing the same geometry, then the method will output equally accurate results, which is a weaker guarantee. It's not clear to me that the encoder architecture will provide exactly the same feature if the input mesh is slightly changed.

2. How well is the method robust to irregular mesh? One major concern I have toward this method is that, if the input mesh is sample in a extremely non-uniform way (which seems to be the case for many physics problem as higher numerical precision will be required in one place than the other), this will require the interpolation to work with different descretization density, which might potentially lead to interpolation error. The experiment regarding irregular mesh is in section 4.2. There is very little description about how the irregular mesh is sampled (or visualization). Also to the bet of my understanding, it's only tested on the irregular mesh, but not using irregular mesh as input. I suspect this also seems to be related to the limitation of the ablation study on interpolation schema, where the difference between the cubic interpolation is not much larer than that of the feature space interpolation. More clarification toward this point will be helpful.
questions: In figure 4(a), is there any intuition why the performance is best at resolution 50? How significant is the the gap between cubic and the proposed method?

Is there any visualization of the input and testing mesh? I think it would be helpful to see some of the visualization on some problem setting to provide better intuition for readers.
limitations: The limitation is properly addressed in section 5.
review_title: null

final_score: rating: 5: Borderline accept: Technically solid paper where reasons to accept outweigh reasons to reject, e.g., limited evaluation. Please use sparingly.
review_content: summary: This paper introduces an architecture for learning mesh-agnostic neural PDE solvers. The architecture consists of encoding, interpolation and forecasting, trained via interpolation loss and forecasting loss. The authors show in a 1D PDE dataset, it outperforms state-of-the-art model MPNN and FNO most of the time in terms of super-resolution on regular and irregular meshes. The authors also investigated how different aspects of the model influences the performance.
strengths_and_weaknesses: Strengths:

The paper tackles an important problem of learning neural PDE solvers which go beyond the resolution it is trained on. The paper is written clearly. The proposed method makes sense.

Weaknesses:

The most important weakness in this paper is its limited evaluation, which only evaluates in 1D PDEs. Since the paper's main contribution is "Mesh Agnostic", we should see experiments where how the method is evaluated in 2D meshes, which is the standard understanding of a mesh. In 1D, a mesh reduces to connected line segments. Compared with 1D, 2D-mesh introduces many new aspects of difficulty, such as more diversity and variations in the shape of the cells, which leads to more complex interpolations, etc. The results in 1D may not generalize to 2D. I would expect having experiments in 2D mesh, both in a regular grid and a irregular mesh, to see how the model compare with SOTA (e.g. using a 2D experiment where FNO or MPNN/MeshGraphNet is evaluated on but evaluated on super-resolution). Also, in 2D where problems are more difficult, the strength of the method compared to baselines may better exhibit.

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Update

The authors have addressed my main concerns in the revision, which add experiments in 2D. In 2D experiments, the MPNN still outperforms the proposed method in the Fig. 15 in all cases, which may not show the relative strength of the proposed method. In light of the above, I have updated my rating from 4 to 5.
questions: No questions.
limitations: The main limitation is indicated in the above weaknesses part.
review_title: null

final_score: rating: 6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
review_content: summary: 1- Summary

The authors propose a PDE solver method based on implicit neural representations address the general limitation of generic numerical PDE solvers that assume the the underlying PDE is discretized and solve it with the Finite Element Method. More specifically, the proposed method lifts such assumptions while increasing prediction accuracy and resolution over generic numerical PDE solvers. The proposed method uses a graph neural network to infer the next state of the system in addition to a physics-based inductive bias in its loss term to constraint the neural network behavior by encouraging it to produce more physically-accurate results. The experiments show that the proposed method is able to estimate spatially continuous solution of the underlying PDE up to 250 frames with physically-consistent accuracy.
strengths_and_weaknesses: 2- Paper strengths

- The proposed PDE solver can generalize to PDEs with higher resolution that PDEs than it was trained on
- The experiments show that the proposed method performs better or similar than SOTA methods

3- Paper weaknesses
Note that some of the weaknesses below are not meant to suggest that the authors should address them during the discussion prior in order to change my opinion. This is because I understand that addressing some of the weaknesses might take a long time or might require significant changes to the proposed method. Although addressing any of these might change my judgment about the work but for the time being the authors may prioritize addressing questions.

- The authors start the paper by talking about climate forecasting as a challenging, high-resolution, problem but their model is far from handling anything related to weather forecasting and their experiments also does not include anything for climate forecasting

4- Additional Comments

Related Work:
- The related work section is written a bit mechanistically and does not highlight some of the distinctive differences of the prior methods in contrast to the authors’ proposed method.

Results:
- It would be good if the authors could provide a summary of the results at the beginning of this section.
questions: - It is not clear how/when the proposed model fails. Have the authors tested their model's limits?
limitations: I am personally hesitant to question societal impacts of any scientific discovery that is not purposefully aiming to promote certain views/directions that directly and objectively harm the society. This is because I believe in the long-term this limits the natural evolution of discovery of the Truth, similar to how protection of the free expression principle leads to democracy. Arguably, any genuine scientific discovery could be eventually used for negative purposes. Therefore, I do not think the reviewers’ subjective opinion should be used in any way as an objective to decide the direction of scientific discoveries.

Regardless, I do not think the proposed method would have any direct negative impact in the society.
review_title: null

Meta-review: metareview: The paper proposes an architecture that maps from an input function to an output function that can handle unstructured meshes. On a set of extensive experiments the effectiveness and robustness of the model is shown. As in other models (like Fourier Neural Operator) the PDE itself is not present in the loss, so 'solution' has to be used with caution. However, the overall design and presentation is impressive, and there is a general agreement between the reviewers about the importance of the work.


 
confidence: Certain
Decision: Accept