Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: 
The paper proposes a method for physics-informed downsampling of the terrain map for scalable inundation modelling.


Main paper contributions are stated to be as following:

- new method for modelling physical processes with deep learning. Specifically, authors combine a deep neural network (DNN) with the partial differential equations (PDEs) describing the flow of water. The DNN downsamples the elevation map, and the downsampled elevation map is fed into the PDEs, where each time step is a recurrent unit, to calculate the water height solution)
-creating of a dataset for a particular problem

main_review: === Summary ====

The paper proposes a method for physics-informed downsampling of the terrain map for scalable inundation modelling.


Main paper contributions are stated to be as following:

- new method for modelling physical processes with deep learning. Specifically, authors combine a deep neural network (DNN) with the partial differential equations (PDEs) describing the flow of water. The DNN downsamples the elevation map, and the downsampled elevation map is fed into the PDEs, where each time step is a recurrent unit, to calculate the water height solution)
-creating of a dataset for a particular problem

== Comparison to previous work ==

Applications of neural network models for solving PDEs is not particularly new (Optimally weighted loss functions for solving PDEs with Neural Networks (2020) R. van der Meer, C. Oosterlee, A. Borovykh).  At the same time, this idea was also used for flood prediction, however in quite different setting (https://arxiv.org/abs/1908.10312). Therefore, current approach, with this formulation and application has not been introduced before. 

==Theory==

Authors discuss downsampling of elevation maps for efficient shallow water flow modelling. This approach is in the intersection of three important field of research: (1) using neural networks for efficient PDE modelling; (2) using deep learning for efficient downsampling of elevation maps; (3) flood modelling. This problem formulation is equally novel and important. 

==Method==

Authors provide detailed explanation of the proposed approach in supplementary material, but I have not seen the code (apologies if it is attached but I have not noticed it). 
The proposed model architecture consists of several models, which together estimate the fluid state. Authors use modified ResNet-18 architecture  as downsampling neural network. 
In the experiments, authors use the proposed neural network modl to solve the shallow water equations on the coarse grid, over variety of elevation maps and boundary conditions. They show that the proposed method is able to provide accurate flood modelling.

=== Pros === 

The proposed method is novel and interesting, authors provide clear description of the method, experiments and outcome, together with the data and code.
They also provide assessment of broader implications and theoretical contribution on the domain of interest. 

===Cons===

The results of current model and comparison to previous work are somewhat scattered in the text. What value this work adds to the traditional methods? 
The model, used in this work, is ResNet, which is quite a ‘classical’ deep learning model, therefore the work does not introduce a particularly novel ML method.

=== Basis for recommendation ===

This paper introduces some important concepts and methods and can be published in NeurIPS 2021. However, some clarification is needed to make this work easier to read.

=== Questions to address in the revision ===

Please state explicitly where the code/data can be found
I would be interested to see more detailed explanation of the contribution of how neural networks are combined with PDEs and ow it compares with previous work (add more detailed explanation to lines 90-114) . How current results can be compared to previous work?
Please state explicitly the results, comparison to previous work and to the baseline (these are somewhat scattered in the text). What value this work adds to the traditional methods? Can you please quantify “significant error reduction” (line 320)?

limitations_and_societal_impact: n/a
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: The paper presents a deep learning-based method for coarse-grain discretization in solving the PDEs of flood modeling. A ResNet is used to downsample the input terrain, and the coarse-grain output is then fed into a numerical solver. A fine-grain numerical solver generates the ground truth for end-to-end training of the downsampling neural network. The work also builds a dataset of elevation maps and their flood simulation results.  The experiment shows that the proposed method surpasses the baseline that uses the average pooling downsampling. 
main_review: Pros:
- The idea of integrating discretization and numerical solver in an end-to-end trainable system seems interesting. 

- The paper is mostly well-written and easy to follow. 

Cons:
- The work lacks clear motivation for developing such a solver-aware downsample learning strategy. As mentioned in the related work, there have been several learning-based coarse-grain discretization methods in the prior work. It is unclear why the previous discretization methods are insufficient for flood modeling. 

- While learning solver-aware downsampling seems to be a sensible design, the paper lacks details on some of its main components: 1) What is the coarse grid representation $\hat{z}$? and what are the different coarse grids formed by the network output? 2) What is the computation graph of the numerical solver, and how the gradient is back-propagated? 3) The simulation iterations are inconsistent in different places: in Line 225, it says one hour of hydraulic flow, while in Line 283, it uses 6 hours. What about using 1-hour short-run in Sec 5.1? 

- The main paper did not include the run-time complexity of training and inference for the overall system. 

- The experimental evaluation is lacking in multiple aspects and the overall claims are not very convincing. First of all, the method is compared with a simple baseline (average pooling) only and misses other more related work such as [4] and [A]. Secondly, it is unclear why the flow accuracy uses the Huber loss, which seems to downplay the large errors. Finally, Figure 5 did not provide detailed quantitative results and those results can only be found in the appendix. For the target application, it is unclear how significant the errors are for the flood modeling.  

[A] Mishra, Siddhartha. “A Machine Learning Framework for Data-Driven Acceleration of Computations of Differential Equations.” Mathematics in Engineering 1.1 (2018): 118–146.

Post rebuttal: The author's rebuttal addressed most of my concerns and I would like to raise my rating to the positive side. 
limitations_and_societal_impact: Seems adequate.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper investigates the problem of downsampling high-resolution elevation maps for more computationally feasible flood modeling simulations. For this purpose, a new pipeline is introduced, that features a downsampling network for the elevation maps, and a differentiable PDE solver for the 2D shallow water equation. The authors also provide their simulated large-scale dataset of flood modeling simulations. Finally, the proposed method is evaluated on 1) an example case that highlights the ability to backpropagate gradients through a large number of simulation steps, 2) an experiment with different boundary conditions, and 3) a generalization experiment to new elevation maps.
main_review: **Originality:** In terms of machine learning, the paper mainly uses existing techniques, however the differentiable implementation of the 2D shallow water equations is interesting. The main novelty is the general approach to the problem. Compared to previous approaches that mainly focus on improving the PDE solver, this paper focuses on improving the input to the solver, which leads to a quite unique perspective. Existing work is properly discussed and even nicely categorized.

**Quality:** The learning setup, loss function, and network architecture are well motivated.
I am not an expert on flood modeling techniques, but I think the experiments are set up thoroughly and produce informative results. Especially, that the network clearly learns to isolate the modified embankment in Section 5.1 over such a large number of simulation and backpropagation steps is impressive. Over the results shown in the main paper and the appendix, the method seems to outperform the intuitive average pooling baseline; not by a huge margin, but quite consistently.

**Clarity:** Overall, the paper is easy to understand, and written and structured well. Only two minor aspects could be described more clearly. First, in Figure 1 a downsampling operation should be added between “True Fluid State” and “Loss Function”, such that the Figure better matches the description in the text. Second, the footnote or the text in Section 4.4 regarding the Huber loss should mention how both map pixel values are combined (this is only clear with the appendix). Otherwise the description in the text mentions *L* as a function of two arguments, while the footnote uses *L* as a function of one argument.

**Significance:** The paper focuses on the rather specific problem of flood modeling, but that in itself already has some direct practical implications and applications as discussed in Section 7. Furthermore, there are some ideas that might be applicable to other PDEs and solvers as well, like the general usage of the differentiable solver within the pipeline, and the perspective of improving inputs to solvers, compared to improving the solver itself.

Overall, the paper provides a novel perspective on flood modeling where deep learning augments a 2D shallow water simulation. In my opinion, this work makes some solid contributions and only has minor weaknesses, leading to my overall evaluation as accept.

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***After rebuttal:*** After reading the authors response and the other reviews, I still think the paper is interesting and only has minor weaknesses. As a result, I see no reason to update my initial evaluation of accept.
limitations_and_societal_impact: Throughout the paper and appendix, limitations and aspects that could be improved upon in future work are sufficiently addressed and clarified. One aspect, that could have been investigated are failure cases where the downsampling network overlooks crucial areas in the elevation map, which the average pooling baseline did capture. I assume such cases would only occur in an out-of-distribution test scenario though.

Apart from such failure cases, I do not see any clear potential negative societal impact of the proposed method that should have been discussed.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: This is an interesting paper in which the authors introduce a new downsampling method for generating coarse elevation maps. A new framework was also built to incorporate the governing PDE (and PDE solver) in the training process. The authors have shown the effectiveness of this proposed method on a dataset with simulated water height at fine resolution. 
main_review: I have a few comments on this paper.
Q1. Although the use of PDE shows a lot of promise, I am wondering whether this brings any negative impact. For example, PDEs may use simplified or approximate representation and thus observed samples (not simulations) may not strictly follow the PDE. If this is also true for flood prediction, the authors may discuss how this issue may affect the proposed method.
Q2. How accurate it is when the numerical solver is used at the coarse resolution? The PDE represents the continuous relationships that underlie the system. If the samples are too coarse, I am concerned that whether the PDE solver (e.g., using Taylor expansion) is still able to produce good simulations since it has no information about the values in between of two points. I totally agree that downsampled data is helpful in saving the time cost for simulation, but just worried about the accuracy. 
Q3. How will other meteorological drivers play a role in flood mapping using this method? It is intuitive that rainfall may be a key factor for the dynamics of the water depth. It would be great to discuss whether the proposed method can also leverage such data sources (e.g., by combining the generated elevation map and other data sources). 
Q4. While the authors have already shown the effectiveness of the proposed algorithm, it would be more convincing if more comparisons can be conducted agaisnt existing machine learning approaches (w/ or w/o using PDEs).

limitations_and_societal_impact: This paper uses deep learning methods for an important problem of flood mapping. I hope to see that this work to be applied to larger regions, e.g., national scale, to aid in flood control. The proposed method can also be potentially used for other applications where downsampling is needed and a governing PDE is given.
review_title: null

Meta-review: The paper proposes a method for physics-informed downsampling of the terrain map for scalable flood modeling. The application problem is extremely important and challenging. The proposed solution, while not totally novel, is reasonable. The experiment would be more convincing if thorough evaluation can be conducted with strong baselines in the fields. We hope that the authors can seriously consider improving the evaluation in the final version of the paper. 
Decision: Accept (Poster)