Venue: ICLR 2020 Conference

Reviews:

final_score: rating: 8: Accept
review_content: Summary:
The paper considers the problem of predicting node and/or edge attributes in a physics-governed system, where we have access to measurement data on discrete sensor locations. In contrast to other methods, they explore the usefulness of a Spatial Difference Layer (SDL) that learns a representation of the gradients on the edges and the Laplacian on the nodes, where the parameters to create those operators are learnable weights. The SDL layer is concatenated with the original graph and fed into a recurrent graph network (RGN) to predict node and/or edge properties. 

Strengths:
- This research is very relevant for physics inspired machine learning, since many physical systems are governed by underlying differential equations.
- The authors show on synthetic data experiments that SDL is capable of representing the derivatives of a physical system.
- Real world use case for temperature prediction is presented with encouraging results.

Weaknesses:
- While comparison to RGN represents a rather strong benchmark, it would be interesting to see a comparison to a graph learning model that is specifically designed for weather forecast.
- Just adding the Spatial Difference Layer using numerical methods (method RGN(StardardOP) and RGN(MeshOP)) can diminish prediction power for a long time horizon. This result suggests that those gradients might not help the prediction.
- The inclusion of an h-hop neighborhood is not quite clear. What value for h was used in the experiments? Is this really necessary, when RGN by itself propagates the signal to neighbors that are further away?

Additional comments:
- 2.1 and 2.2: On first reading, it's a bit confusing why there are two different equations for (∆f)i. The motivation of the second equation should be made more explicit.
- 3.1 lacks explanation of what is train / test set, which is given only in the appendix. This is critical information to understand the use cases of the model and should definitely be in the main body.
- In 3.2 formatting of a(i), b(i) and c(i) is confusing. Why is the setup only similar to Long et al? It would be nice to point out the differences and explain why it wasn't exactly the same.
- 4.1: Was the train/validation/test split done in contiguous segments? I.e. are the 8 months of training data January to August? How is the problem of learning different seasons handled?

review_title: Official Blind Review #1

final_score: rating: 8: Accept
review_content: The authors propose new architectures that simulate difference operators. According to my experience, this research is important since PDEs are the most commonly used form to represent known physical relationships in dynamical systems.  The proposed method has novelty in that it uses advanced machine learning architectures to estimate physically meaningful variables. The authors further investigate how to use the proposed spatial difference layer in two tasks. 
I would suggest improving this research on these aspects: 
1.	The proposed method should be evaluated on more datasets. The difference information is used in almost any real-world dynamical systems and thus it would be more convincing to show the effectiveness on diverse applications, e.g., object tracking, the variation of energy and mass across space and time.
2.	It would be interesting to design a test scenario where governing PDEs are known. Is it possible for your method to uncover the relationship of gradients that govern the system?
3.	How sparse the data is? It would be better to have an experiment where data is intentionally hidden to control the sparsity and then evaluate the performance.
4.	A side question: In real-world systems, the observations are not only governed by PDE, but also unknown noisy factors, missing physics, etc. Can your method handle the noisy data/labels?
5.	The proposed method has some complex components. I would encourage releasing the code and the simulated dataset upon acceptance. 

review_title: Official Blind Review #3

final_score: rating: 6: Weak Accept
review_content: This paper proposes a method to reduce numerical error when predicting sequences governed by physical dynamics. The idea is that most physics simulators use finite difference differential operators, and the paper adds trainable parameters to them (the derivative and Laplacian operators). The added parameters are computed by a GNN. The paper concats the original graph feature and the output of the differential operators, and inputs them to a recurrent GNN to obtain the final prediction. 

I think the idea is interesting. Incorporating modulated derivative and Laplacian operators into physical simulators is novel and well justified. It could strengthen the argument is there is more justification of why this particular parameterization is selected. 

I think the experimental evaluation is somewhat adequate. There are a good selection of baselines including both manually designed iterators and GNNs. In particular, the weather prediction experiment show improved performance over several baselines. I am not familiar with this task or its state-of-the-art performance, but I am convinced that the proposed approach is superior compared to the claimed baselines (RGN, GRU). 

I have several confusions or concerns about the synthetic experiments

1. In the synthetic experiments, is the evaluation task different from the training task? It is unclear from the description how well the learned parameters generalize. Does the method generalize to a. New functions/dynamics b. New graphs with similar properties (e.g. another graph draw from the same distribution) c. New graph with different properties (e.g. more or less sparse)? 

2. One short-coming of the synthetic experiment is the lack of error bars, or analysis of statistical significance. I think some of the improvements are not large enough to be statistically convincing without additional analysis. It seems necessary to experiment on multiple random problems (e.g. with random meshing, dynamics parameters). 

Minor comments:

A related idea is “Learning Neural PDE Solvers with Guarantees” which modulates the finite difference iterative solver with deep networks, but the objective is solving PDEs with known dynamics instead of prediction with unknown dynamics. Conversely, the method the authors proposed seem also useful for speeding up PDE solvers. 

I think there is an error in the type definition of f and F in section 2.1. The two claimed types contradict each other.

review_title: Official Blind Review #2

Meta-review: All reviewers agree that this research is novel and well carried out, so this is a clear accept. Please ensure that the final version reflect the reviewer comments and the new information provided during the rebuttal
Decision: Accept (Poster)