Multi-scale hydraulic graph neural networks for flood modelling

Bentivoglio, Roberto; Isufi, Elvin; Jonkman, Sebastiaan Nicolas; Taormina, Riccardo

doi:https://doi.org/10.5194/egusphere-2024-2621

Preprints

https://doi.org/10.5194/egusphere-2024-2621

Preprints

20 Sep 2024

| 20 Sep 2024

Multi-scale hydraulic graph neural networks for flood modelling

Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, and Riccardo Taormina

Abstract. Deep learning-based surrogate models represent a powerful alternative to numerical models for speeding up flood mapping while preserving accuracy. In particular, solutions based on hydraulic-based graph neural networks (SWE-GNN) enable transferability to domains not used for training and allow including physical constraints. However, these models are limited due to four main aspects. First, they cannot model rapid differences in flow propagation speeds; secondly, they can face instabilities during training when using a large number of layers, needed for effective modelling; third, they cannot accommodate time-varying boundary conditions; and fourth, they require initial conditions from a numerical solver. To address these issues, we propose a multi-scale hydraulic-based graph neural network (mSWE-GNN) that models the flood at different resolutions and propagation speeds. We include time-varying boundary conditions via ghost cells, which enforce the solution at the domain's boundary and drop the need of a numerical solver for the initial conditions. To improve generalization over unseen meshes and reduce the data demand, we use invariance principles and make the inputs independent from coordinates' rotations. Numerical results on dike-breach floods show that the model predicts the full spatio-temporal simulation of the flood over unseen irregular meshes, topographies, and time-varying boundary conditions, with mean absolute errors in time of 0.05 m for water depths and 0.003 m² s⁻¹ for unit discharges. We further corroborate the mSWE-GNN in a realistic case study in The Netherlands and show generalization capabilities with only one fine-tuning sample, with mean absolute errors of 0.12 m for water depth, critical success index for a water depth threshold of 0.05 m of 87.68 %, and speed-ups of over 700 times. Overall, the approach opens up several avenues for probabilistic analyses of realistic configurations and flood scenarios.

Received: 20 Aug 2024 – Discussion started: 20 Sep 2024

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Journal article(s) based on this preprint

23 Jan 2025

Multi-scale hydraulic graph neural networks for flood modelling

Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, and Riccardo Taormina

Nat. Hazards Earth Syst. Sci., 25, 335–351, https://doi.org/10.5194/nhess-25-335-2025,https://doi.org/10.5194/nhess-25-335-2025, 2025

Short summary

Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, and Riccardo Taormina

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2621', Anonymous Referee #1, 27 Sep 2024

This paper presents a deep-learning based approach for 2D surface flood modelling that builds on a previously existing model, SWE-GNN. It proposes an alternative model that overcomes the need of a numerical solver to determine initial conditions and includes other novelties to improve the speed and generalization of the model. The paper demonstrates that the model can benefit from fine-tuning to generalize to new case studies. In my opinion, the research is interesting and is very well presented. I am therefore recommending the paper for publication, after minor revisions.

General comments
The paper proposes a model for flood modelling and is applied only to a specific case of floods; dike-breach floods. It would be nice to have a section in the discussion on how the model could apply to other kind of floods, such as fluvial and pluvial floods for example.

Specific comment
L84 – L85: For reproducibility, I would suggest clarifying how a mesh is classified as being too small and how the resolution of the fine mesh is chosen.
L93: It might be better to define what epsilon is here rather than in L104.
Eq. 3 and 4: I think that it might be useful to have the clarification of what h_di and h_si are.
L190: I would suggest replacing ‘training simulations’ by ‘training data’ here for clarity.
L193: You should explain that water level is the sum of the water depth and elevation of the cell as this is might not be straightforward.
Section 2.5: Clarify that O is the output dimension.
Table 1: To which resolution of the mesh does the edge length refer?
Section 3.1: I believe a concise explanation of how the Manning’s n coefficients are defined for the synthetic dataset would be helpful. I’m curious to know if these coefficients are spatially variable as the model might face challenges in extracting meaningful insights from them if they were spatially uniform.
L270-275: You provide a nice and clear explanation of CSI. However, you consider 2 thresholds (0.05 m and 0.3 m) while you explain in L273-275 that the flooded and non-flooded cells are distinguished from another in TP, FP and FN. However, this isn’t entirely accurate as cells with water depths below 0.3 m could still be considered as flooded. I would try to be more accurate and write for example: ‘TP are true positives, i.e. number of cells where both model and simulations predict water levels above the threshold value’. The same applies for the description of FP and FN.
L285: I would suggest adding a sentence stating that you are assuming the enhanced SWE-GNN also outperforms the other models, given that the original SWE-GNN outperforms them and despite the added modifications.
L305: Do you know why the MAE of h increases? Could it be due to error propagation throughout the simulation? If you have any insights into the cause of this increase, it might be worth adding it.
Fig. 9 and Fig. 10: Consider using different color scales as it is difficult to visualize the differences in water levels as it is. Also, could you clarify which mesh (e.g. finest mesh) is shown in the figures?
Fig. 10: It is really nice that you detail what the negative and positive values correspond to in the captions of Fig. 9 and Fig. 10. However, I would recommend revising the caption in Fig. 10 to enhance clarity, e.g. ‘positive values indicate that the model estimates later arrival times than the numerical simulation, while negative values indicate that the model predicts earlier arrival times’
Technical corrections
Throughout the manuscript (e.g., L14, L51, L223, caption of Fig.5): Uncapitalize ‘the’ in ‘The Netherlands’
L178: ‘via edges directed towards’ rather than ‘via directed edges towards’
L292: Clarify ‘is comparatively faster than the numerical model’
Throughout the manuscript: Add a spacing between the numbers and the ‘m’ for meters
L302-303 and 305: Replace the four ‘in correspondence of’ with ‘occur at’, ‘occur simultaneously to’, ‘near’ and ‘at’ respectively
Fig. 8: You might want to keep the notation of CSI_0.05 and CSI_0.3 like in the manuscript, i.e. add the ‘m’
Table 2 and 3: Write the units as [10-2 m] and [10-2 m2/s]
L330: Typo in ‘dataset’
L365: Typo in ‘dependent’
L393: ‘analyze’ rather than ‘analyse’ to keep consistency with the previous sentence
L466: Remove ‘the’ in ‘comes from the an increase’

Citation: https://doi.org/10.5194/egusphere-2024-2621-RC1
- AC1: 'Reply on RC1', Roberto Bentivoglio, 30 Oct 2024
  
  We thank the Reviewer for the helpful comments and suggestions. We address them individually in the attached document.
  
  Citation: https://doi.org/10.5194/egusphere-2024-2621-AC1
RC2:
'Comment on egusphere-2024-2621', Julian Hofmann, 03 Nov 2024
Review of "Multi-Scale Hydraulic Graph Neural Networks for Flood Modelling"
This paper presents a multi-scale, hydraulic-based graph neural network model (mSWE-GNN) designed to enhance flood modelling. Building on previous models like SWE-GNN, the authors introduce novel elements such as rotation-invariant inputs, ghost cells for boundary conditions, and a multi-scale architecture to simulate flood events over variable topographies and unseen meshes. The paper includes extensive theoretical explanations and benchmarking evidcence that demonstrate the model's effectiveness across different flood scenarios. Overall, this research is timely and relevant, contributing meaningfully to the field of flood modeling with machine learning. I support the publication of this paper following some revisions.
General Comments
The model presented in this paper, mSWE-GNN, demonstrates impressive capabilities in simulating dike-breach flood scenarios specifically. However, its applicability appears limited to this particular type of flood. Given the diversity of flood types—such as fluvial and pluvial floods—extending the model's capabilities or discussing potential adaptations for these other scenarios would significantly enhance its relevance and utility. A broader application across flood types would open up further opportunities for practical deployment and highlight the robustness of the proposed multi-scale architecture in different hydraulic contexts. Including a discussion on how the model might be adapted for other flood types would be a valuable addition.
However, the paper limits its application to dike-breach floods and does not explore other flood scenarios. Adding a section discussing potential adaptations for other flood types, such as fluvial or pluvial floods, would strengthen the paper’s relevance. Additionally, the inclusion of a mass conservation term in the loss function, though discussed in the appendix, warrants further investigation, particularly regarding its influence on model stability.
Specific Comments
L89-92: For clarity and reproducibility, it would be beneficial for the authors to provide additional detail on the criteria for mesh size. Specifically, a brief explanation of how mesh resolution is chosen and classified as "too small" would enhance reproducibility.

Eq. 3 & Eq. 4: It may be helpful to specify definitions for hdih_{di}hdi and hsih_{si}hsi, as these variables are key in understanding the propagation rule and its application.

Boundary Condition Section (2.3): This section introduces ghost cells effectively but lacks explicit examples or visual representation of how they are implemented for various boundary types (e.g., inflows and outflows). Including this could aid in comprehending the technique's generalizability to different boundary configurations.

Fine-Tuning Evaluation (Section 4.2): While the fine-tuning approach is well-motivated, it would be helpful to elaborate on the potential for overfitting given that a single simulation is used for fine-tuning and testing. The authors might explore this limitation and offer suggestions for future validation.

Discussion on Physical Constraints (Appendix B): The addition of mass conservation loss is an interesting consideration. However, it would be helpful if the authors could provide further analysis or reasoning on why this term, although theoretically sound, did not yield a consistent improvement in testing. This section could benefit from a comparison of other physical constraint approaches in neural networks.

Figures and Tables
Figures 9 and 10: The flood arrival time and water depth predictions presented are informative, but some color scales may not adequately differentiate between close water depth levels. Adjusting these color scales would make the figures more accessible and highlight differences more clearly.

Table 2: To support interpretability, clarify which mesh resolution is referred to in this table. Also, in Table 3, presenting units consistently as [10−2 m][10^{-2} \text{ m}][10−2 m] and [10−2 m2/s][10^{-2} \text{ m}^2/\text{s}][10−2 m2/s] improves clarity and aligns with standard formatting practices.

Appendix Figure A1: This figure is valuable in understanding unit discharge modeling, but it may benefit from a clearer representation of errors by using varying scales or alternative visual indicators for over- and under-predictions.

Technical Corrections
L302 & L305: The phrase “in correspondence of” could be replaced with more direct alternatives such as "occur at" or "align with" for clarity.

Throughout manuscript: Maintain consistent formatting of units, particularly with spacing between numbers and "m" for meters.

L466: “comes from the an increase” should be corrected to remove the redundant article “the.”

Conclusion
In summary, the paper presents an interesting deep learning model with clear advancements over existing approaches in flood modeling. The mSWE-GNN’s ability to generalize and efficiently process multi-scale data is convincingly demonstrated. Nevertheless, a limitation of the current model is its focus on simulating dike-breach floods alone. Given the range of flood types—such as fluvial and pluvial floods—future work extending this model’s application to other types would significantly increase its value and utility. Including a discussion on potential adaptations for different flood scenarios would further enrich the paper, offering insights into the generalizability and versatility of the multi-scale approach.
Citation: https://doi.org/10.5194/egusphere-2024-2621-RC2
- AC2: 'Reply on RC2', Roberto Bentivoglio, 05 Nov 2024
  
  We thank the Reviewer for the helpful comments and suggestions. We have addressed them individually in the attached document.
  
  Citation: https://doi.org/10.5194/egusphere-2024-2621-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2024-2621', Anonymous Referee #1, 27 Sep 2024

This paper presents a deep-learning based approach for 2D surface flood modelling that builds on a previously existing model, SWE-GNN. It proposes an alternative model that overcomes the need of a numerical solver to determine initial conditions and includes other novelties to improve the speed and generalization of the model. The paper demonstrates that the model can benefit from fine-tuning to generalize to new case studies. In my opinion, the research is interesting and is very well presented. I am therefore recommending the paper for publication, after minor revisions.

General comments
The paper proposes a model for flood modelling and is applied only to a specific case of floods; dike-breach floods. It would be nice to have a section in the discussion on how the model could apply to other kind of floods, such as fluvial and pluvial floods for example.

Specific comment
L84 – L85: For reproducibility, I would suggest clarifying how a mesh is classified as being too small and how the resolution of the fine mesh is chosen.
L93: It might be better to define what epsilon is here rather than in L104.
Eq. 3 and 4: I think that it might be useful to have the clarification of what h_di and h_si are.
L190: I would suggest replacing ‘training simulations’ by ‘training data’ here for clarity.
L193: You should explain that water level is the sum of the water depth and elevation of the cell as this is might not be straightforward.
Section 2.5: Clarify that O is the output dimension.
Table 1: To which resolution of the mesh does the edge length refer?
Section 3.1: I believe a concise explanation of how the Manning’s n coefficients are defined for the synthetic dataset would be helpful. I’m curious to know if these coefficients are spatially variable as the model might face challenges in extracting meaningful insights from them if they were spatially uniform.
L270-275: You provide a nice and clear explanation of CSI. However, you consider 2 thresholds (0.05 m and 0.3 m) while you explain in L273-275 that the flooded and non-flooded cells are distinguished from another in TP, FP and FN. However, this isn’t entirely accurate as cells with water depths below 0.3 m could still be considered as flooded. I would try to be more accurate and write for example: ‘TP are true positives, i.e. number of cells where both model and simulations predict water levels above the threshold value’. The same applies for the description of FP and FN.
L285: I would suggest adding a sentence stating that you are assuming the enhanced SWE-GNN also outperforms the other models, given that the original SWE-GNN outperforms them and despite the added modifications.
L305: Do you know why the MAE of h increases? Could it be due to error propagation throughout the simulation? If you have any insights into the cause of this increase, it might be worth adding it.
Fig. 9 and Fig. 10: Consider using different color scales as it is difficult to visualize the differences in water levels as it is. Also, could you clarify which mesh (e.g. finest mesh) is shown in the figures?
Fig. 10: It is really nice that you detail what the negative and positive values correspond to in the captions of Fig. 9 and Fig. 10. However, I would recommend revising the caption in Fig. 10 to enhance clarity, e.g. ‘positive values indicate that the model estimates later arrival times than the numerical simulation, while negative values indicate that the model predicts earlier arrival times’
Technical corrections
Throughout the manuscript (e.g., L14, L51, L223, caption of Fig.5): Uncapitalize ‘the’ in ‘The Netherlands’
L178: ‘via edges directed towards’ rather than ‘via directed edges towards’
L292: Clarify ‘is comparatively faster than the numerical model’
Throughout the manuscript: Add a spacing between the numbers and the ‘m’ for meters
L302-303 and 305: Replace the four ‘in correspondence of’ with ‘occur at’, ‘occur simultaneously to’, ‘near’ and ‘at’ respectively
Fig. 8: You might want to keep the notation of CSI_0.05 and CSI_0.3 like in the manuscript, i.e. add the ‘m’
Table 2 and 3: Write the units as [10-2 m] and [10-2 m2/s]
L330: Typo in ‘dataset’
L365: Typo in ‘dependent’
L393: ‘analyze’ rather than ‘analyse’ to keep consistency with the previous sentence
L466: Remove ‘the’ in ‘comes from the an increase’

Citation: https://doi.org/10.5194/egusphere-2024-2621-RC1
- AC1: 'Reply on RC1', Roberto Bentivoglio, 30 Oct 2024
  
  We thank the Reviewer for the helpful comments and suggestions. We address them individually in the attached document.
  
  Citation: https://doi.org/10.5194/egusphere-2024-2621-AC1
RC2:
'Comment on egusphere-2024-2621', Julian Hofmann, 03 Nov 2024
Review of "Multi-Scale Hydraulic Graph Neural Networks for Flood Modelling"
This paper presents a multi-scale, hydraulic-based graph neural network model (mSWE-GNN) designed to enhance flood modelling. Building on previous models like SWE-GNN, the authors introduce novel elements such as rotation-invariant inputs, ghost cells for boundary conditions, and a multi-scale architecture to simulate flood events over variable topographies and unseen meshes. The paper includes extensive theoretical explanations and benchmarking evidcence that demonstrate the model's effectiveness across different flood scenarios. Overall, this research is timely and relevant, contributing meaningfully to the field of flood modeling with machine learning. I support the publication of this paper following some revisions.
General Comments
The model presented in this paper, mSWE-GNN, demonstrates impressive capabilities in simulating dike-breach flood scenarios specifically. However, its applicability appears limited to this particular type of flood. Given the diversity of flood types—such as fluvial and pluvial floods—extending the model's capabilities or discussing potential adaptations for these other scenarios would significantly enhance its relevance and utility. A broader application across flood types would open up further opportunities for practical deployment and highlight the robustness of the proposed multi-scale architecture in different hydraulic contexts. Including a discussion on how the model might be adapted for other flood types would be a valuable addition.
However, the paper limits its application to dike-breach floods and does not explore other flood scenarios. Adding a section discussing potential adaptations for other flood types, such as fluvial or pluvial floods, would strengthen the paper’s relevance. Additionally, the inclusion of a mass conservation term in the loss function, though discussed in the appendix, warrants further investigation, particularly regarding its influence on model stability.
Specific Comments
L89-92: For clarity and reproducibility, it would be beneficial for the authors to provide additional detail on the criteria for mesh size. Specifically, a brief explanation of how mesh resolution is chosen and classified as "too small" would enhance reproducibility.

Eq. 3 & Eq. 4: It may be helpful to specify definitions for hdih_{di}hdi and hsih_{si}hsi, as these variables are key in understanding the propagation rule and its application.

Boundary Condition Section (2.3): This section introduces ghost cells effectively but lacks explicit examples or visual representation of how they are implemented for various boundary types (e.g., inflows and outflows). Including this could aid in comprehending the technique's generalizability to different boundary configurations.

Fine-Tuning Evaluation (Section 4.2): While the fine-tuning approach is well-motivated, it would be helpful to elaborate on the potential for overfitting given that a single simulation is used for fine-tuning and testing. The authors might explore this limitation and offer suggestions for future validation.

Discussion on Physical Constraints (Appendix B): The addition of mass conservation loss is an interesting consideration. However, it would be helpful if the authors could provide further analysis or reasoning on why this term, although theoretically sound, did not yield a consistent improvement in testing. This section could benefit from a comparison of other physical constraint approaches in neural networks.

Figures and Tables
Figures 9 and 10: The flood arrival time and water depth predictions presented are informative, but some color scales may not adequately differentiate between close water depth levels. Adjusting these color scales would make the figures more accessible and highlight differences more clearly.

Table 2: To support interpretability, clarify which mesh resolution is referred to in this table. Also, in Table 3, presenting units consistently as [10−2 m][10^{-2} \text{ m}][10−2 m] and [10−2 m2/s][10^{-2} \text{ m}^2/\text{s}][10−2 m2/s] improves clarity and aligns with standard formatting practices.

Appendix Figure A1: This figure is valuable in understanding unit discharge modeling, but it may benefit from a clearer representation of errors by using varying scales or alternative visual indicators for over- and under-predictions.

Technical Corrections
L302 & L305: The phrase “in correspondence of” could be replaced with more direct alternatives such as "occur at" or "align with" for clarity.

Throughout manuscript: Maintain consistent formatting of units, particularly with spacing between numbers and "m" for meters.

L466: “comes from the an increase” should be corrected to remove the redundant article “the.”

Conclusion
In summary, the paper presents an interesting deep learning model with clear advancements over existing approaches in flood modeling. The mSWE-GNN’s ability to generalize and efficiently process multi-scale data is convincingly demonstrated. Nevertheless, a limitation of the current model is its focus on simulating dike-breach floods alone. Given the range of flood types—such as fluvial and pluvial floods—future work extending this model’s application to other types would significantly increase its value and utility. Including a discussion on potential adaptations for different flood scenarios would further enrich the paper, offering insights into the generalizability and versatility of the multi-scale approach.
Citation: https://doi.org/10.5194/egusphere-2024-2621-RC2
- AC2: 'Reply on RC2', Roberto Bentivoglio, 05 Nov 2024
  
  We thank the Reviewer for the helpful comments and suggestions. We have addressed them individually in the attached document.
  
  Citation: https://doi.org/10.5194/egusphere-2024-2621-AC2

Peer review completion

AR: Author's response | RR: Referee report | ED: Editor decision | EF: Editorial file upload

ED: Publish subject to minor revisions (review by editor) (19 Nov 2024) by Kai Schröter

AR by Roberto Bentivoglio on behalf of the Authors (20 Nov 2024) Author's response Author's tracked changes Manuscript

ED: Publish as is (25 Nov 2024) by Kai Schröter

AR by Roberto Bentivoglio on behalf of the Authors (26 Nov 2024) Manuscript

Journal article(s) based on this preprint

23 Jan 2025

Multi-scale hydraulic graph neural networks for flood modelling

Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, and Riccardo Taormina

Nat. Hazards Earth Syst. Sci., 25, 335–351, https://doi.org/10.5194/nhess-25-335-2025,https://doi.org/10.5194/nhess-25-335-2025, 2025

Short summary

Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, and Riccardo Taormina

Data sets

Raw datasets for paper "Multi-scale hydraulic graph neural networks for flood modelling" Roberto Bentivoglio https://doi.org/10.5281/zenodo.13326595

Roberto Bentivoglio, Elvin Isufi, Sebastiaan Nicolas Jonkman, and Riccardo Taormina

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Short summary

Deep learning methods are increasingly used as surrogates for spatio-temporal flood models but struggle with generalization and speed. Here, we propose a multi-resolution approach using graph neural networks that predicts dike breach floods across different meshes, topographies, and boundary conditions with high accuracy and up to 1000x speedups. The model also generalizes to larger, more complex case studies with just one additional simulation for fine-tuning.


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