Physics-Aware Downsampling with Deep Learning for Scalable Flood Modeling
Authors: Niv Giladi
Zvika Ben-Haim
Sella Nevo
Yossi Matias
Daniel Soudry
Venue: NeurIPS 2021 Poster (2021)
Keywords: Partial differential equation
Inundation modeling
Physics-informed
Downsampling
Flood modeling

Abstract: Background. Floods are the most common natural disaster in the world, affecting the lives of hundreds of millions. Flood forecasting is therefore a vitally important endeavor, typically achieved using physical water flow simulations, which rely on accurate terrain elevation maps. However, such simulations, based on solving partial differential equations, are computationally prohibitive on a large scale. This scalability issue is commonly alleviated using a coarse grid representation of the elevation map, though this representation may distort crucial terrain details, leading to significant inaccuracies in the simulation.\\
Contributions. We train a deep neural network to perform physics-informed downsampling of the terrain map: we optimize the coarse grid representation of the terrain maps, so that the flood prediction will match the fine grid solution. For the learning process to succeed, we configure a dataset specifically for this task. We demonstrate that with this method, it is possible to achieve a significant reduction in computational cost, while maintaining an accurate solution. A reference implementation accompanies the paper as well as documentation and code for dataset reproduction.

Physics-Aware Downsampling with Deep Learning for Scalable Flood Modeling
Niv Giladi Zvika Ben-Haim'
Sella Nevo!
Yossi Matias'
Daniel Soudry²
Google Research
Technion - Israel Institute of Technology
(giladiniv, daniel. soudry))aaa
{zvika, sellanevo, pgasilegoogle.com
Abstract
Background. Floods are the most common natural disaster in the world, affecting the lives of hundreds of millions. Flood forecasting is therefore a vitally important endeavor, typically achieved using physical water flow simulations, which rely accurate terrain elevation maps. However, such simulations, based solving partial differential equations, are computationally prohibitive on large scale. This scalability issue commonly alleviated using coarse grid representation of the elevation map, though this representation may distort crucial terrain details, leading to sigmificant inaccuracies in the simulation.
Contributions. We train a deep neural network to perform physics-informed sampling of the terrain map: we optimize the coarse grid representation the terrain maps, so that the flood prediction will match the fine grid solution. For the learning process succeed, we configure a dataset specifically for this task, We demonstrate that with this method, it is possible to achieve a significant reduction in computational cost, while maintaining an accurate solution. A reference imple mentation accompanies the paper as well as documentation and code for dataset reproduction.
informed down-
Introduction
A physical process is time-evolving phenomenon marked by gradual transitions through series of states, following physical rules. Scientists attempt to model these physical processes by employing analytical descriptions that correspond to the underlying physics of these processes. Commonly, the analytical descriptions, or physical models, such as conservation laws, are in the form of partial differential equations (PDEs). Physical models the backbone behind modeling phenomena such as weather forecasts, flood forecasting, aerodynamics, chemical processes, and more. Traditionally, these physical models are solved numerically, problems. One typical numerical method is to discretize the PDEs using Taylor expansion and repeatedly integrate time over short intervals. However, solving PDEs numerically on large scale is challenging because of the need to resolve spatiotemporal features over many scales.
closed form solution is available for most
In this paper, focus hydraulic models, which widely used physical models. The goal of these models is to simulate the flow of water in a given environment. We are specifically interested in simulating water flow through floodplain in order to predict inundation. i.c., where and to what extent will flooding occur. Specifically, inundation modeling we are given an elevation map of the terrain, boundary conditions, initial conditions and wish predict, the water height in each pixel at some future time with maximum accuracy and minimum computational cost. Such prediction is an important task in operational flood alert systems [ 43 [0]
35th Conference on Neural Information Processing Systems (NcurIPS 2021).

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Unfortunately, inundation modeling is often computationally intractable when modeling flow on large scale with fine grid resolution. A coarse grid representation of the elevation map can significantly improve computation time, for two reasons. First, less computation is done at each step of the PDE solver. Specifically, in discretized PDE the per-step spatial computation done between neighboring pixels. Therefore, if we downsample d-dimensional map by a factor of we decrease this per-step computation by a factor of x Second, the maximum stable time step in the numerical solution can be increased by a factor of z, entailed by the FriranttFriedrichs-- Taking both factors into account, we gain a total theoretical speedup of ad+1, For example, this paper we will show that for the wvo-dimensional problem (d 2) of inundation modeling, retain fairly good accuracy when downsampling by a factor of 16, leading to a significant 4096 speedup.
(CFL) condition [23] 77.
So far it was considered challenging to coarse- -raiin a general elevation map without losing accuracy in the inundation prediction. In particular, small details in the elevation map can significantly impact the water flow evolution. Narrow canals, dams, and embankments (barriers preventing a river from flooding an area) are examples of such details that require special care. There are several typical methods to represent representation, as small, important details might get lost [Н] 44 43 For example, when applying standard downsampling methods, 8 meter wide embankment might be reduced to half its original height above its neighboring surroundings in a 16 meters resolution coarse map. More complex downsampling techniques can be employed, but in practice, to achieve high-quality simulations at a coarse resolution, domain expertise is required, along with a deep understating of the underlying physics of the specific problem.
elevation map
coarse grid, all of which can cause a sub-optimal
To address these limitations, we optimize the coarse grid representation of the elevation map so that its water height solution will match the accurate fine grid solution. This is accomplished by combining a deep neural network (DNN) with the PDEs describing the flow of water. The DNN downsamples the fine grid 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. Thus, the DNN is optimized directly on the water height solution, through the PDEs. This also requires back-propagating through the PDEs, whose time-steps need to be unrolled. We demonstrate that this approach can yield a better coarse grid representation than conventional methods, in terms of water flow estimation. To this end, we configured a dataset designed specifically for inundation modeling. Each sample consists of a fine grid elevation map, boundary conditions, and simulation time. The ground truth of each sample is the corresponding fluid state, calculated on a fine grid. This dataset is available along with code for further use.
The contributions of our work include:
A new framework for learning how to coarse- coarse-grain PDE together with its external environment by minimizing the distance between the solutions on the fine and coarse grids (Sec. 4.1).
novel configured dataset for 2D water flow estimation, with accessible code for reproduction
(Sec.
A demonstration that gradients backpropgated for tens of thousands of steps through the PDE remain informative (Sec. 5.1 and can pinpoint key features (Fig. 2
A demonstration of the accuracy of our approach in inundation modeling, for two different application settings (Sec. an2 Since we decrease the map resolution by a factor 16, this entails 4096x speedup. Such speedup has significant impact on the feasibility of real-time flood warning for large populations (Sec. 8).
2 Related Work
The integration of machine learning with physical models, described by PDEs, has emerged in recent years useful tool for efficiently solving computational science problems. In particular, the use of machine learning models promising in problems where little is known about the underlying process, or in problems where the computational overhead is significant 42]
Data driven models. Sharifi et al. [32), Vandal et al. 40), Gentine et al. 14. and Srivastava et al. 36 use basic ML methods to interpolate between coarse and fine representations to improve accuracy
'The code for this paper is available at http:///hhcccommccc...smssssssss

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in modeling. In these studies, they use supervised data driven models that predict the solution of the physical model. Data driven models can fail to capture important complex relationships between physical variables directly from data, and it is necessary to ensure that the learned models follow physical laws.
Physics constrained loss function. To force the predictive models to follow physical laws, a common technique is to incorporate the physical constraints as additional terms in the loss function 29 30
DNNs have several advantages. First, the computation of the physical loss term can be done without labeled data. In addition, the physical loss term acts as a regularizer. so it can reduce the search space of the optimization. These models has shown success in several tasks such as forward solving of PDEs and inverse problems [42]. However, physical loss loss.
119
35 4. These physics-informet
terms require fine-tuning to find the optimal weighting between the predictive loss and the physical
Physics guided architecture. Another common methodology is physics guided architecture, where domain knowledge is utilized design architectures that capture physics dependencies among variables [33
12 9 17 24J. These models are more interpretable than regular models. However, the architecture design can be useful for specific type of PDEs, and make no sense for others. Designing a specific architecture for each physical model can be a tedious task of its own.
Data driven discretization. Perhaps the most closely related works showed that modeling features of the fluid dynamics and numerical scheme with neural networks can enable solving PDEs on much coarser grids than is possible with standard numerical methods 4 15 21 38]. By learning to modify the PDE solver, these works enable using a coarser grid (potentially with some overhead) while retaining accuracy and numerical stability. However, they did not examine how to optimally coarse grain the external environment (given as input to the solver) - as we do here, for terrain elevation maps in the context of flood modeling. In this case, it is critical that the coarse-grained map retains the relevant features, to prevent drastic differences between fine and coarse solutions (e.g.. floods). In other words, these previous works address the discretization error entailed in numerical modeling, while this paper suggests a method to reduce the input error, such as errors in the elevation map. The two approaches (modifying the PDE solver vs. modifying the external environment) are orthogonal, and can be potentially combined in future work. Empirically, we found that the input error is dominant in our setting of flood modeling, however, it is an interesting question to quantify this more precisely. Note our approach has no computational overhead during inference since we use the same solver, just on a coarser grid with a modified map. Evidently, we provide experiments on a significantly larger scale than these previous works.
3 Physical Model: The Shallow Water Equations
In this section we describe the physical model of fluid flow used in this paper, i.e. the hydraulic model. At the most fundamental level, fluid flow is described by the Navier-Stokes equations. In practice, solving these equations directly is often infeasible, and simpler variations are derived for specific problem settings. One such variation is the shallow water equations [22], a set of partial differential equations that describe water flow when water depth is much smaller than horizontal extent, as is generally the case for floods. The shallow water equations are derived by depth-integrating the Navier-Stokes equations 41 g 77. These assumptions allow a considerable simplification in the mathematical formulation and numerical solution, which enables analysis on much larger horizontal length scales.
To write the shallow water equations, we first define the following notation: is the time index, (x. y) the spatial horizontal coordinates, ha,, y;t) is the water height relative to the terrain elevation
à 2(2., v). q (g (a, y; t), 92 (z. y; t)) is the horizontal mass flux of the fluid, V 0zx ay) is the horizontal gradient operator, n is Manning's friction coeffieient, and g is the gravitational acceleration constant. The shallow water equations are described by two conservation laws. First, the conservation of mass, which gives rise to the continuity equation.
Ōh
V --
Øt
Second, the conservation of momentum, which yields the momentum equation,
dq
ot + ghV (h + +2) + gn2 ||q]]
h7/3 q = 0.
(2)
3

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Note that eqs. and 2 neglect advection and the Coriolis force, as is common for flood modeling. We follow De Almeida & Bates 7 numerically approximate eqs. and 2 from the continuous domain into the discrete domain, where the numerical solution uses finite difference scheme applied to staggered grid. In the discretized domain, functions becomes arrays (e.g., (xx, y) becomes 2).
In real-world flood forecasting settings, data for the elevation map would be collected ahead of time, and the boundary conditions for the model would be provided either by real-time measurements or hydrologic forecasts. One would then run the hydraulic model (eqs. [ and 2 in real-time, or offline on a wide range of possible boundary conditions. The simulation provides both the flood extent and the inundation depth at every pixel. This information can then be used to inform both authorities working on mitigation and response efforts, as well as individuals that may be affected by the expected flooding (e.g., alert apps on mobile phones). However, to do this, the simulation must be done on a large scale and fast enough to be practically relevant. Next, we show how this can be done using physics-aware downsampling of the elevation map.
4 Learning to Downsample the Elevation Map
In this section, we formulate the map downsampling problem for several application settings and define the training setup. Then, we describe data generation and configuration. Lastly, we describe the architecture considerations and optimization settings which enable the learning process success.
4.1 Problem Setup
Consider a downsampling DNN fw and numerical water flow solver S with fixed target time t. The downsampling network is a deep neural network parameterized by weights W. transforms a two dimensional fine grid elevation map E Rm into a coarse grid elevation map 2 E IR R via the mapping& 2 fw(z), The fine and coarse grid water heights, h E Rm and h E k respectively, are computed by the numerical solver with initial and boundary conditions e such that
h = S(z, h c)2.
A loss function ((h, h) penalizes any difference between the fine grid water height h and the coarse grid water height h. The fine grid water height is downsampled to coarse grid for calculation of the loss. Note that this differs from calculating the water height on a coarse grid, since the water height is calculated on fine grid and only then is downsampled to coarse grid. Since we are interested in inundation modeling in coarse resolution, the deviations coarsening the fine water height are acceptable and can be accomplished by simple averaging of the fine water height. Figure I depicts deepicts the system overview. Our first general goal is to find a mapping fw through the minimization problem:
W' arg arg min Ez,c [((z, c), S(fw(z),
W
The second goal we consider is a weaker variation of eq. (3) which is relevant in operational flood forecasting systems, described in Sec. 5.2 In such a setting, the fine grid elevation map is known, and we wish to find a coarse representation of the elevation map that will generalize to many boundary conditions. We condition the expectation
and the minimization problem can be rewritten
W (z) = arg min E. [e (5(z, c), S(fw(z() c) |z].
(4)
To train and evaluate model based on the minimization problem in eq. data is required, which we next discuss how to configure.
3 training and validation
4.2 Dataset configuration
Elevation map curation. In recent years, improvements in remote sensing and data processing at large scales have improved the access to reliable, diverse, and rich earth-science data. case in point is the extensive collection of 1-meter Digital Elevation Models (DEMs) published by the United States Geological Survey (USGS) M. These DEMs are created using airborne lidar in --mete x 1-meter cell size resolution, and cover most of the United States, including flood-prone areas around the Mississippi River and the Arkansas River K 27.. We based our dataset on terrain data from the floodplains of those rivers in three different locations. These areas a mere fraction of the available

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S
h
C
True Fluid State
Numerical Solver
l
Input Boundary Initial Conditions
S
h
Loss Function
Z
fw Downsampling
Numerical Solver
Estimated Fluid State
Input Fine Grid Terrain
Coarse Grid Terrain
Network
Figure 1: Training setup for the downsampling neural network fw. Each data sample consists of fine grid elevation map z, with boundary and initial conditions Two solutions calculated - the fine grid water height h and the coarse grid water height h. The latter uses coarse grid version of the input elevation map å. We train the downsampling neural network fw. to minimize the loss measuring the distance between the two solutions (eqs.
This is done by backpropagating the loss gradients through the numerical solver.
USGS data, but focused them due compute resource limitations. The data was further processed into tiled elevation maps. Mountainous terrains and areas with missing data were filtered to be appropriate for the hydraulic simulation. Each elevation map contains x000 pixels, covering a 2km 2km region. We found this size is large enough to capture the way differences between coarse and fine grid solutions can affect flooding even distant locations. We gathered total of 5183 elevation maps. For each elevation map, we defined boundary conditions as influx, outflux and discharge. More details about data configuration can be found in Sec. A. of the appendix.
Ground truth calculation. For the scope of this paper, we wish to calculate the flow of the water, ie., the water height at each pixel, over each data sample, given its boundary conditions over some time t. It possible to construct the corresponding ground truth by simulating the water flow over the fine grid elevation maps. For each elevation map, its ground truth is 2000 matrix, describing the water height at each pixel. This is computationally demanding task, but can be performed offline once. We provide access to the full dataset along with code reproduce the data and expand to more data samples.
4.3 Architecture
We use the ResNet-18 architecture [66 as the core of our DNN model. The last layer of the classifier was removed, and 1 convolution was used instead to transform the last hidden layer with multiple channels, into a single channel output. In our experiments, we use a downsampling factor of 16x, which is natively implemented in the ResNet- 18 model. For smaller downsampling factor possible to remove all layers with resolution lower than needed and apply the x 1 convolution for single channel output. is also possible that other models can perform as well, but these kinds of experiments are beyond the scope of this paper.
For the output scheme, we first notice that simple average coarsening techniques works reasonably well for many of the elevation maps and for a large portion of the spatial domain. This is because fine important features effecting the hydraulic flow relatively scarce. Therefore. we suggest that the downsampling DNN fw output will consists summation of an average pooling coarse grid elevation map and a correction map given by the ResNet-18 model. This is implemented by adding an average pooling skip connection from the input of the ResNet- 18, directly to its output. This has two main advantages. First, the ResNet- 18 output values have a smaller range and variance, in comparison to the full coarse grid elevation maps. Second, the starting point of the DNN accuracy is close to the accuracy of average pooling downsampling. Both advantages help the learning process converge significantly faster and make performance less sensitive to initialization where the DNN outputs the entire coarse grid elevation map rather than a correction (details in the Appendix).
in comparison to the case
5

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Data normalization. It is common practice to normalize the input data of the DNN, e.g. by performing standardization, min-max normalization, and more. However, using multiplicative normalization such as standardization will distort the vertical scale of the elevation map and essentially remove important information. Indeed, two elevation maps can have the same normalized form, but encompass different hydraulic properties, and possibly different optimized coarse grid representation. Consequently, we normalize the input elevation map only by subtracting its mean.
4.4
Optimization scheme
The per-sample loss is of the per-pixel loss on all the map pixels Eis E(hi,g hag). Here & Huber loss which exhibited good convergence behavior. The training loss was optimized using Adam [20], and batch size of 32 samples.
There are several challenges training the DNN in the setting described Sec. 4.1 First, to obtain gradients for each sample, one must calculate the hydraulic solution, i.e., iteratively applying the PDEs in the forward pass, and then calculate the gradients in the backward pass, through the PDEs. The backward calculation through the PDFs takes the majority of the iteration time. Ideally, we would like to reach a steady state solution, where the velocity field and total mass do not change. However, to keep the training computationally feasible, stop the solver after it finished simulating one hour of hydraulic flow. This still might require thousands of recurrent iterations of the PDEs, depending the CFL condition described in Sec.
Gradient checkpointing. As discussed in Sec. we define each fine grid elevation map to be relatively large, 2000 x 2000 pixels, to allow long-distance flow effects to manifest. As result, the activation maps are large, causing memory bottleneck and limiting the batch size that can be used in each node. In addition, the flow of water calculation might take thousands of recurrent iterations, as described in Sec. [ increasing both memory and compute time. To address this computational challenge, we use the gradient checkpointing mechanism, described in Siskind & Pearlmutter [34] and Chen et al. [6]. The checkpointing is applicd both to the DNN, and to the numerical solver.
Numerical stability of the PDEs. In addition to instabilities originating in DNN non-convex optimization, one needs to take into account numerical instability of the PDEs when performing a learning rate scan. The numerical scheme of the Shallow Water Equations is prone to numerical instability issues, much like other numerical schemes. This instability more apparent when there are large variations in the terrain, causing rapid changes in the water velocity and depth. When training with learning rates larger than 10-2 with Adam, and 10-1 with Stochastic Gradient Descent (SGD), divergence problems occur, especially at the beginning of training: over-correction by the gradients cause topographic exaggeration by the model which leads to numerical instability of the PDEs. To avoid this, we performed our learning rate scan with in the relatively low range of [10-3 3 10 100110p to find the empirically optimal learning rate in terms of convergence rate and generalization, as well as numerical stability of the PDEs. Adam achieved better generalization and therefore is used.
Experiments
In this section, we leverage our physics-aware DNN to accurately solve the shallow water equations on coarse grid, over variety of elevation maps and boundary conditions. Specifically, we:
1. Demonstrate that the system (Sec. 4.1 map, using the accumulated gradient after backpropagation through the PDEs.
can locate hydraulically significant details of the elevation
2. Optimize the DNN over a single elevation map with multiple boundary conditions (eq. and demonstrate better performance compared traditional downsampling. This is typical setting in operational flood modeling systems, where repeated simulations are performed with different boundary conditions.
3. Train the DNN over a training set of different elevation maps and boundary conditions (eq. and show generalization capabilities on both elevation maps and boundary conditions. This would enable fast downsamping of maps in new areas.
c.. f(h, h) 0.5(h - h)3 if lA hl < and E(h, h) = -h 0.5 otherwise.

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Throughout the experiments, we use an averaging baseline for comparison. This baseline is intuitive for terrain elevation map downsampling, and is being used in traditional inundation modeling studies We also consider slightly less naive downsampling techniques, and we demonstrate that they are not superior to the baseline we use (Appendix AA2.3). More details on the experiments can be found in Appendix A.2
11 43
5.1
Backpropagation through PDEs
In this section, we demonstrate that hydraulically meaningful details in the elevation map are identified by backpropagation through the PDEs, allowing the model to learn to preserve these details when downsampling. Specifically, show that the gradient signal can properly flow even through tens of thousands of unrolled recurrent units representing the PDE time-steps. We explicitly construct an example to demonstrate that important details in the elevation map can be detected by the gradients: We start with an elevation map with an evident embankment, and artificially flatten a few pixels of that embankment. The modified elevation map is shown in Fig. 2a with a red square around the flattened pixels (a zoomed-in version of the red square is shown in Fig. (bHtop). From the boundary conditions we used, water flows in from the right bottom part of the elevation map and flows out from right upper part. With the flattened pixels, water can flow through the embankment, inundating a large area left of the embankment which not inundated in the non-modified elevation map. This difference is shown in Fig. 2d where red pixels are more inundated on the modified elevation map solution (over-flooding). and blue pixels are more inundated on the non-modified elevation map solution (under-flor (ooder-flooding) In turn, the loss is high because of the large inundated area. A meaningful gradient will identify the discrepancy as resulting from the flattened embankment pixels, rather than the entire inundated region. The resulting gradient map of the elevation map after backpropagation is shown in Fig. 2c] As can be seen, the gradient is highest at the flattened pixels, especially compared to the inundated region, indicating that those few pixels contributed significantly to the deviation from the non-mon modified elevation map solution. We emphasize that was required to unroll more than ten thousand iterations to compute the gradient, we simulated 6 hours of hydraulic flow.
(a)
(b)
(c)
(d)
04
02
-04
06
08
Figure 2: Gradient flow through PDEs. (a) A modified elevation map with a few pixels of the embankment flattened.. (d) Difference map between the solution calculated on the modified elevation map and the non- modified elevation map solution. (c) Gradient magnitude of the modified elevation map, computed by back- propagating through the PDEs. (b) Zoom-in on the modified pixels of the elevation map (top) and on the gradient magnitude map of the same location (bottom).
5.2 Generalization over multiple boundary conditions
In this section, we find an optimized coarse grid representation of a single elevation map with varying boundary conditions, as the minimization problem in Eq. (4). To do so, we optimize DNN where each data sample uses the same elevation map, but with different boundary conditions - including influx and outflux locations as well as discharge. This setting is useful for repeated simulations with different boundary conditions over the same spatial domain, a typical setting in operational flood modeling systems [5]. We validate the model on randomly sampled boundary conditions. The baseline we compare ourselves against remains average pooling. In Figure we show a comparison between the water flow solution on fine grid elevation map and each of the solutions calculated on coarse grid elevation maps, both for the trained DNN and baseline. As can be seen, solutions calculated on the DNN coarse grid elevation map are closer to the fine grid solution. We provide more examples different elevation maps in Appendix A.2.4
7

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In Figure 4 we provide example of visible features preserved by the DNN, On the right image, there a small embankment that preserved entirely in the DNN version, while in the baseline version the embankment is averaged with its surroundings, causing it to almost disappear. On the left image, there narrow canal with embankments on either side. Again, the DNN preserves both the canal and most of the embankments, while in the baseline version, the canal is averaged with the embankments, causing partial flattening of the terrain at the canal location. emphasize that the DNN coarse grid elevation maps outputs of two different models that were trained each image separately.
LOSS: 1.31e-02
Loss: 1.04e-03
Loss: 2.57e-03
Loss: 7.77e-03
meter
Baseline
0.5
00
Loss: 9, 9.64e-04
Loss: 6.12e-04
Loss: 8.95e-04
Loss: 2. 2.65e-00
DNN
meter
Figure 3: Generalization over different boundary conditions. Leftmost images coarse grid elevation maps downsampled using DNN (bottom) and average pooling baseline (top), where the DNN trained on different boundary conditions and same elevation map. Each column comparison solutions between the DNN and the baseline elevation maps, for different boundary conditions. Each image on those columns is a difference map between the coarse and fine grid solutions along with the Huber loss. Red indicates pixels inundated in the coarse grid solution, and blue indicates pixcls more inundated in the fine grid solution. The color scale, meters, appears on the right. Solutions calculated on the DNN elevation map achieve better accuracy compared to the baseline.
1.0
0.5
00
100
Figure 4: Capturing fine details important to water flow, Two patches taken from different elevation maps. The two larger images are the fine grid patches and the two smaller images the left of cach fine grid image the coarse grid patches of the average pooling baseline (top) and the DNN (bottom). Note the preservation of hydraulically significant features such as canals and embankments in the DNN as opposed to the bascline.
5.3 Generalization over different elevation maps
For scalable inundation modeling, robust model for coarse grid representation required, which is able to generalize to elevation maps and boundary conditions unseen during training. We train such model on 4031 elevation maps, each with different boundary conditions. We validate the model on 1 1155 unseen elevation maps, each with different boundary conditions. We compare the accuracy of the water flow solutions obtained on coarse grid between the DNN elevation maps and the average pooling elevation maps. The metric used for water flow solution accuracy is Huber loss. We used Adam with a learning rate 10-3 and batch size of samples. The training was done 90 epochs. Figure 5a shows the convergence plot of the model and its improvement over the baseline. Note that

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the model starts with a loss similar to the baseline, because of the architecture scheme described in section
Next, we plot the 2D histogram of the DNN's loss compared to the baseline (Figure 5b). The resulting distribution is almost entirely below the linear dashed line, indicating better accuracy by the DNN, matching Figure [bb] There is a large number of data points with small loss (both by the model and the DNN), meaning that for a large portion of the data, average pooling achieves reasonable accuracy. Overall, the modified elevation maps exhibited an improvement of the Huber loss by a factor of 2 over all the tested elevation maps. This improvement increases to factor of 2.5 as we look only harder examples (90% percentile). Other accuracy metrics show similar results with larger improvement factor of up 3. Additional analysis is provided in Appendix A.2255
(b)
+00
Train
0.07
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0.06-
10a
Easeline
0.05
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0.04
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10
0.02
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20
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0.01 0.02 0.03 0.04 0.05 0.06 0.07
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Figure 5: Generalization over different elevation maps. (a) Convergence plot of the model loss for both training and validation. (b) A comparison of the per-sample Huber loss values of the baseline (x-axis) and the DNN (y-axis). The color scale is logarithmic.
DNN model surpass the accuracy of the baseline, with an improvement of the Huber loss by factor of 2. Additional analysis is provided in Appendix A.2.5
6 Summary
Enabling accurate inundation modeling a vitally important endeavor for flood forecasting, and physical water flow simulations are typically core component of that endeavor. These simulations rely on accurate elevation maps, and computationally demanding on a large scale. Elevation map coarsening can significantly reduce the computation required, but hydraulically important fine details might get distorted on a coarse grid. We tackle this problem by training a physics aware DNN that downsamples elevation maps to a coarse grid representation. while preserving details important for hydraulic simulation. We do so by incorporating a physical model into the learning process, such that gradients calculated through the PDEs. We also configure a dataset specifically for this task and make it available for further use. We consider two applicable training settings for operational flood forecasting systems, and demonstrate that such training is feasible and that the learned DNN outputs meaningful elevation maps with better water flow results compared to prevalent methods.
7 Broader impact
Floods cause thousands of fatalities, affect hundreds of millions of people, and cost approximately $10 billion every year [10 39 Flood early warning systems have the potential to significantly reduce both fatalities and economic costs, in many cases between 30-50% 31 25 28). Inundation models are a critical component of such systems, as many people will not take protective action without information that is spatially accurate [5]. However, the vast majority of fatalities and harms occur in low and middle income countries, where both financial resources and expertise support such efforts scarce [10 26). As a result, the requirement of enormous computational resources, or the expertise to intelligently downscale elevation maps to enable more efficient simulation, can be infeasible for the most important uses of these models. This work can bypass need for either these constraints, providing tool for the resource-constraine
to use in their modeling, or helping support efforts to scale up global inundation models, such as those pursued by organized by Google, Fathom, ECMWE, CGIAR and others.

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8 Acknowledgments
We thank Elad Hoffer for technical advising, as well as Yaniv Blumenfeld and Mor Shpigel Nacson for valuable comments the manuscript. The research of DS was supported by the Isracl Science Foundation (grant No. 1308/18), and by the Israel Innovation Authority (the Avatar Consortium).
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12


Reviews:
[{"review_id": "y6PLOwdLbPB", "reviewer": "Reviewer_R5nW", "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.", "questions": "", "limitations": "", "rating": 7, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "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.\n\n**Quality:** The learning setup, loss function, and network architecture are well motivated.\nI 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.\n\n**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.\n\n**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.\n\nOverall, 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.\n\n------------------------------------------------------------------------------------------------------------------------------------\n***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.", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "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.\n\nApart from such failure cases, I do not see any clear potential negative societal impact of the proposed method that should have been discussed. ", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "4 hours"}, "extra_text": {}}, {"review_id": "wCnbmJfSz5q", "reviewer": "Reviewer_EGwq", "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. ", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "Pros:\n- The idea of integrating discretization and numerical solver in an end-to-end trainable system seems interesting. \n\n- The paper is mostly well-written and easy to follow. \n\nCons:\n- 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. \n\n- 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? \n\n- The main paper did not include the run-time complexity of training and inference for the overall system. \n\n- 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.  \n\n[A] Mishra, Siddhartha. “A Machine Learning Framework for Data-Driven Acceleration of Computations of Differential Equations.” Mathematics in Engineering 1.1 (2018): 118–146.\n\nPost rebuttal: The author's rebuttal addressed most of my concerns and I would like to raise my rating to the positive side. ", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "Seems adequate.", "review_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "hubk26hU38j", "reviewer": "Reviewer_Vnr6", "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. ", "questions": "", "limitations": "", "rating": 7, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "I have a few comments on this paper.\nQ1. 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.\nQ2. 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. \nQ3. 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). \nQ4. 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).\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "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_text": "", "extra_scores": {"needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2"}, "extra_text": {}}, {"review_id": "3WRWjlVucp", "reviewer": "Reviewer_48GU", "summary": "\nThe paper proposes a method for physics-informed downsampling of the terrain map for scalable inundation modelling.\n\n\nMain paper contributions are stated to be as following:\n\n- 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)\n-creating of a dataset for a particular problem\n", "questions": "", "limitations": "", "rating": 7, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "=== Summary ====\n\nThe paper proposes a method for physics-informed downsampling of the terrain map for scalable inundation modelling.\n\n\nMain paper contributions are stated to be as following:\n\n- 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)\n-creating of a dataset for a particular problem\n\n== Comparison to previous work ==\n\nApplications 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. \n\n==Theory==\n\nAuthors 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. \n\n==Method==\n\nAuthors 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). \nThe proposed model architecture consists of several models, which together estimate the fluid state. Authors use modified ResNet-18 architecture  as downsampling neural network. \nIn 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.\n\n=== Pros === \n\nThe proposed method is novel and interesting, authors provide clear description of the method, experiments and outcome, together with the data and code.\nThey also provide assessment of broader implications and theoretical contribution on the domain of interest. \n\n===Cons===\n\nThe results of current model and comparison to previous work are somewhat scattered in the text. What value this work adds to the traditional methods? \nThe 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.\n\n=== Basis for recommendation ===\n\nThis 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.\n\n=== Questions to address in the revision ===\n\nPlease state explicitly where the code/data can be found\nI 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?\nPlease 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)?\n", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "n/a", "review_text": "", "extra_scores": {"ethical_concerns": "n/a", "needs_ethics_review": "No", "ethics_review_area": "", "time_spent_reviewing": "2 days"}, "extra_text": {}}]

Author rebuttal:

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)