FLOOD SIMULATION WITH PHYSICS-INFORMED MESSAGE PASSING

We thank the reviewer for the comments. We were able to address each one of them and hope that the reviewer would consider upgrading the rating based on our responses

1. The GNN-based model, including the message-passing mechanism, is not described very thoroughly.
   We have rewritten the Method section providing a detailed description of our architecture.

2. Does it then indicate that it is this aspect, and not the model itself, which has most impact? If that is the case, then shouldn't we be trying to incorporate the water retention aspect in the existing architectures rather than proposing a new architecture?
   The design of the message passing plays a key role in the dispersion phase. The GCN and GAT models considered in our work are implemented the same way as CommGNN with the retention component but with the message-passing of the vanilla GCN and GAT, respectively. We show that they do not perform as good as CommGNN because their message-passing is not as expressive as the one we propose following the conservation of momentum and mass

3. We have no results on how the proposed approach compares with the hydrological/hydrodynamic models used by domain scientists for flood simulation
   The training data is generated using the hydrodynamical model LISFLOOD-FP which we use as the ground truth. There is not enough spatio-temporal data readily available to train an ML model. Therefore, ML models are trained on the simulation output of hydrodynamic models so that they can match their accuracy.

4)Fig 1 shows how water from two adjoining nodes at higher elevation can flow into one node at lower elevation. But how can water from a higher region divide itself into two lower locations?
We apply the D8 flow direction graph, which is represented by linking each node to its steepest neighbor. Thus, for each node, there is only one outgoing edge (Page 4 & Appendix A.3). The D8 flow direction has long been used by domain experts in hydrology especially when dealing with very shallow water. However, we recognize that this becomes a limitation for later stages (which is shown in our experiments in Table 2). In future work, we will investigate a better graph representation such as a dynamic graph which will be determined based on the current surface elevation and potential energy surface

5. Is elevation the only factor that influences the dispersion of water? What can be other factors and can they be taken into account in the proposed model? Soil infiltration and friction can be other factors that can be taken into account. However, these data are not readily available, especially solid infiltration extraction which is an active area of research. The incorporation of these features will result in a more involved shallow water equations and therefore will require some modification to the message passing proposed in our current work.

6. Do the competing models take DEM or any other auxilliary information as input? We use the same input, that is, DEM and precipitation, for all the models for a fair comparison.

7. Is it possible to have an experimental result that shows the water retention and dispersion stages separately?
   CommGNN$^-$ considers only the water dispersion. MLP, used as one of the baselines, does not have propagation. It can be considered as a retention-only method

8. Can the retention abilities be different in different regions? Which factor in the model takes the retention capacity into account?
   Yes, water retention can be different over a region because the precipitation data is spatially distributed, resulting in different amounts of rainwater falling in different places of a region. Other local attributes could also be incorporated as inputs in the retention phase.

9. In the experiment, has it been assumed that each region is of uniform elevation? What is the spatial resolution of the DEM, rainfall and flood water depth map?
   No, they are not uniform. We are using real-world data representing watersheds with various topographies. Details about the regions are shown in Appendix A.4, Table 5, and Figure 6. The resolution of each DEM grid cell is 30m by 30m, and so are the rainfall and water depth maps.

We thank the reviewer for the comments. We were able to address each one of them and hope that the reviewer would consider upgrading the rating based on our responses.

1. The reason for using the proposed GNN instead of a conventional numerical solver are not clear. It seems not to be the goal to improve the accuracy, since the model is trained with simulated flooding data. Is it accelerating the simulation? Runtime comparisons with numerical solvers are not included in the paper
   Previous works (Bates, 2022) have shown that ML models are faster, but they do not perform as accurately as conventional models. The current focus is to produce ML models that can perform as good as conventional models, and hopefully better. The reason why we are using flood simulation data is because there is not enough true spatiotemporal flooding data that can be used for training an ML model.

2. Solely from a deep-learning viewpoint, the contributions are not significant. Another "species" of message passing is added to an already vast range of options
   The novelty of our method includes (i) a two-stage paradigm in the spatiotemporal simulation of flooding when accounting for precipitation, which as external sources are rarely considered in current ML simulations of dynamic systems. (ii) The design of a message passing that follows the conservation of mass and momentum (iii) Applications on real-world regions, which have complex topologies compared to nicely shaped manifolds considered in current ML for dynamic systems. Through extensive experiments, we show that our proposed method is better than the existing alternatives. We have written the Related Work and Methods sections present our work more clearly.

3. The term "physics-informed" used in the title is not well chosen in my opinion
   Thank you for the suggestion. We will change the title to “Flood Simulation With Physics-guided Message Passing" when the openreview system allows it.

4. The use of a single out-going edge per node assumes that the information is only propagated downhill. This may not be true for later-stage flood simulations
   We agree with the reviewers. This is the main point we showed in Table 2 and pointed out this limitation to address in future work. In future work, we will investigate a better graph representation such as a dynamic graph which will be determined based on the current surface elevation and potential energy surface. That said, a single out-going edge per node seems to work well at the early stages of simulations, and it has long been used by domain experts in hydrology when the water depth is not very high. We show these results comparing the performance between the single edge out-going edge graph and when the entire grid is used as a graph in Table 4 (Appendix A.3).

5. In these two articles, the flooding models account for past states as opposed to the model proposed here. The authors should discuss if that is needed or not
   Thank you for suggesting these works. In our case, based on the design of the model we tried to use past history of precipitation data in our model but we did not notice any performance improvement. However, we cannot give a definitive answer to whether the past history is relevant or not, since we might need to investigate a little bit further to figure out what feature might be suitable with past history.

6. More baselines We have compared our method to MeshGraphNet from the same authors of (Battaglia et al. 2016), (Pfaff et al. 2021), and (Sanchez-Gonzalez et al. 2020). We have added two more baselines MP-PDE, and ConvLSTM (https://proceedings.neurips.cc/paper/2015/file/07563a3fe3bbe7e3ba84431ad9d055af-Paper.pdf) The original versions of these baselines did not perform well. When fine-tuning them and trying out different activation functions and data normalization techniques, we found that log-transformation of the data and tanh activations boosted the performance of these methods. We therefore applied the same transformation to all other methods (including ours) for fair comparison and noticed an improvement, the results of the current version of the paper are much better than the previous ones for all the methods. Our proposed method still performs the best compared to these additional baselines

7. Use ReLU activation after the last layer?
   This was our initial implementation, but we found that this was a little bit difficult to train, probably due to zero gradients, the “dying ReLU” phenomenon.

8. What is the number of parameters of each model?
   We have added the parameters of each model in Table 7 Appendix (A.5)

9. Other research has considered back-propagating through multiple evaluations of the model. Could this be beneficial for the flood forecasting and long-term stability?
   We are not sure we understand. However, this looks like an interesting idea. We would be grateful if the reviewer could share some references.

We thank the reviewer for the comments. We were able to address each one of them and hope that the reviewer would consider upgrading the rating based on our responses.

1. For methods such as Forward Euler to converge there is a CFL condition which bounds ∆t by ∆x and 1 is too large.
   We agree with the reviewer. This was not clear in the paper. Data is generated from numerical hydrodynamic model LISFLOOD-FP, an optimized and specialized model developed by domain experts. At fixed intervals, it outputs snapshots of water depths. However, internally it uses an adaptive time-stepping to ensure convergence. $\Delta t$ here refers to the output time (which remains the same for the entire dataset), not the internal time step used by LISFLOOD-FP. We set $\Delta t$=1 in the current version of our work as it doesn't change

2. Forward Euler not numerically stable, advanced schemes such as RK4 may perform better. We mentioned Foward Euler to show how $\frac{q^t-q^{t-1}}{\Delta t}$ comes from $\frac{\partial q}{\partial t}$ in Eq 8.This is only used for our model, true data is generated by LISFLOOD-FP with an adaptive time-stepping.

3. Numerical solutions as additional baseline Given the complexity and size of the domain, the shallow water equations for flood simulation are solved numerically with optimized numerical hydrodynamic solvers (e.g, LISFLOOD-FP) developed by domain experts. Raw numerical solvers such as FD, FEM can break in some scenarios due to the ground topography inequalities and external boundary conditions such as precipitations.

4. Highly related to MeshGraphNets Our method is different from MeshGraphNet (i) We are dealing with system of PDEs, in MeshGraph only 1 PDE equation is used at a time (ii) Our method has 2 stages: water retention, and water dispersion. This paradigm turns out to be useful for capturing external input such as precipitation. MeshGraphNet does not model such phenomena (iii) We have 2 message-passings that approximate spatial derivative in Eq1&Eq2 with Eq9&Eq8, respectively (iv) The design of these message-passings captures the topography of the domain and the amount of water based on conservation of mass and momentum. MeshagraphNet simply concatenates all the features.(v) We are working on domains with very complex topographies where the continuity of the water elevation can break. MeshGraphNet has been applied only on smooth manifolds.
   We compared our method to MeshGraphNet with same features as our method and showed that our approach achieves better results.

5. Loss function hard-coded The loss function is for training purposes only, with no theoretical derivation. We found that most methods benefited from this loss, especially MeshGraphNet, with an increase of up to 20% in performance. Whereas $L_2$ and $L_1$ would require fine-tuning the learning and decay function for each individual model, which could still perform worse than this loss function.

6. Generalization While we are looking at extending our work to other physical processes, the current version of this work is specific to shallow-water equations in the context of flood simulation. We have improved the motivation of our work in the introduction.

7. No theoretical or convergence properties We have rewritten the method section by providing a justification for the design proposed.

8. More baselines We have added MeshgGraphNet, MP-PDE, and ConvLSTM (Shi 2015) as baselines. However, we had trouble fine-tuning DINO, PINNs, and FourierNet on our dataset because of memory constraints. The original versions of MeshGraphNets and MP-PDEs did not perform well. When fine-tuning them and trying out different activation functions and data normalization techniques, we found that log-transformation of the data and tanh activations boosted the performance of these methods. We therefore applied the same transformation to all other methods (including ours) for fair comparison and noticed an improvement, the results of the current version of the paper are much better. Our proposed method still performs the best compared to these additional baselines.

9. Reason why GNNs not been explored for flood simulations?. Flood simulation poses the following challenges (i) The complexity of the ground topography, which can break the continuity of flows. Even though methods such as MeshGraphNets and DINO have simulated the Navier-Stokes, their experiments are mainly conducted on simpler and smoother manifolds which ensure the continuity of the physical process. (ii) The sparsity of the data: The data is usually very sparse, which makes it difficult to train. (iii) Swift transition of states from dry to wet or wet to dry. Assuming a discretized domain, it is easy to predict water increment on a cell that has already water in it. This explains the good performance of all the methods in Table 2. However, predicting the transition from a dry cell to a wet cell is usually harder (see Table 1).

We thank the reviewer for the comments. We were able to address each one of them and hope that the reviewer would consider upgrading the rating based on our responses.

1. Novelty of the ComGNN model.
   We have expanded the method section by providing a detailed description of the model. In the revised version of the manuscript, we show that our method operates in two stages. Each cell first retains water from the rain and then propagates it following Eq1 & Eq2 Propagation starts with the conservation of momentum (Eq2) where we use a message passing based on the topology of the ground elevation and the amount of retained water (of each cell/node) as an approximation of the spatial derivative to obtain Eq8. This produces a flow that is used in the conservation of mass (Eq1) approximated by Eq10 to output the final water depth. Therefore, the novelty of our method includes (i) a two-stage paradigm in the spatiotemporal simulation of flooding when accounting for precipitation data, which as external sources are rarely considered in current ML simulations of dynamic systems. (ii) The design of a message passing that follows the conservation of mass and momentum (iii) Applications on real-world regions, which have complex topologies compared to simpler manifolds considered in current ML for dynamic systems

2. Eq 1 to 4 appears to describe the application of learnable multilayer
   We have expanded the method section by showing that Eq 8 & 10 (in the current version) are the building blocks of a message passing architecture inspired by the conservation of mass and momentum.

3. Figures in page 8 and 9 could move them in to appendix
   Thank you for the suggestion, we have moved most figures to the appendix and have expanded the method section.

4. The current formulation of the problem may be overly simplistic for readers not well-versed in the field. It would be advisable to include the mathematical model within this section to enhance comprehension
   Thank you for the suggestion. We have included a description of the physics behind a flooding process in section 3 as follows. "The theoretical framework for flood modeling is based on fluid mechanics described by the 3D Navier-Stokes equation. In practice, however, the characteristic vertical length scale of the flow is very small with respect to the characteristic horizontal length scale, resulting in a constant horizontal velocity field throughout the depth of the fluid. The dynamics of a flooding process are, therefore, derived by depth integrating the 3D Navier-Stokes equation, leading to a system of non-linear PDEs called shallow-water equations..."