the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 07 Jul 2025
Hybrid Physics-AI and Neural ODE Approaches for Spatially Distributed Hydrological Modeling
Abstract. Empirical models are among the earliest hydrological models and have evolved from the unit hydrograph to deep learning models. Despite their success, purely data-driven methods often lack interpretability and are highly sensitive to data quality, limiting their generalizability in data-scarce regions or under changing environmental conditions. Conceptual models, traditionally relying on simplified representations of physical processes governed by conservation laws of mass, momentum, and energy, remain widely used in operational hydrology due to their explainability and practical applicability. However, these process-based models inherently face structural uncertainties and a lack of scale-relevant theories—challenges that emerging artificial intelligence (AI) techniques may help address. Moreover, high-resolution models are crucial for predicting extreme events characterized by strong variability and short duration, making spatially distributed hybrid modeling critical in the current context. We introduce a hybrid physics-AI approach that integrates neural ordinary differential equations (ODEs), solved by an implicit numerical scheme, into a spatialized, regionalizable, and differentiable process-based model. The hydrological module is built on a continuous state-space system and an integrated process-parameterization neural network. This hybrid system solves the ODEs governing reservoir dynamics, while embedding a neural network to refine internal water fluxes, all without relying on an analytical solution, instead computing the model states simultaneously. This work also presents an upgraded version of the smash platform following its initial release, featuring a more comprehensive evaluation of hybrid models at relatively fine resolutions of kilometric spatial and hourly temporal scales. The results show that hybrid approaches demonstrate consistently strong and stable performance in calibration and various validation scenarios. Additionally, the neural ODE structure exhibits a hybridization effect that modifies state dynamics and runoff flow, achieving more reliable streamflow simulations for flood modeling.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Journal article(s) based on this preprint
Interactive discussion
Status: closed
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CEC1: 'Comment on egusphere-2025-2797', Juan Antonio Añel, 25 Jul 2025
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AC1: 'Reply on CEC1', Ngo Nghi Truyen Huynh, 25 Jul 2025
Dear Dr. Juan A. Añel,
Thank you very much for your comment regarding the output data. Please find below the Zenodo deposit containing the output results and the scripts used to perform our study with SMASH:
https://doi.org/10.5281/zenodo.16419642This deposit includes scripts to perform calibration/validation and the output files containing the calibrated models/parameters, simulated responses, and metric scores for all 9 models compared in our paper. A detailed description of the contents is provided in the README file of the deposit.
And yes we will include in any future version of our manuscript this new repository in the data and code availability section.
Thank you again for posting your comment.
Best regards,
Ngo Nghi Truyen Huynh, on behalf of the authors
Citation: https://doi.org/10.5194/egusphere-2025-2797-AC1 -
CEC2: 'Reply on AC1', Juan Antonio Añel, 28 Jul 2025
Dear authors,
Many thanks for addressing the mentioned issue so quickly. We can consider the current version of your manuscript in compliance with the Code and Data Policy of the journal.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2025-2797-CEC2
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CEC2: 'Reply on AC1', Juan Antonio Añel, 28 Jul 2025
-
AC1: 'Reply on CEC1', Ngo Nghi Truyen Huynh, 25 Jul 2025
-
RC1: 'Comment on egusphere-2025-2797', Anonymous Referee #1, 23 Aug 2025
Summary
The manuscript proposes a hybrid physics–AI framework for spatially distributed rainfall–runoff modeling. It builds on a differentiable, continuous state-space GR-type hydrological core with 1D routing based on a D8 drainage scheme, augmented by two neural components: (i) a regionalization network that maps physical descriptors to spatially varying hydrological parameters, and (ii) a flux-correction network that adjusts internal model fluxes using meteorological inputs and prior model states. Because state evolution is governed by ODEs, the flux-correction component is trained jointly with an implicit ODE solver in a neural-ODE fashion. Across case studies, the hybrid configurations deliver more accurate and stable streamflow predictions than the classical GR baseline. Overall, this is a timely and valuable contribution to robust, differentiable coupling of process-based hydrology with machine learning.
Major comments- Self-contained presentation and background: The current manuscript would benefit from a concise background on rainfall–runoff modeling (empirical/black-box, conceptual, and distributed approaches; recent ML-based advances and their limitations) to situate the contribution and clarify what is new versus prior work.
- High-level system overview: Given the number of interacting components (regionalization network, flux-correction network, ODE state evolution, PDE/routing), please add a clear schematic that shows data flow for the different configurations listed in Table 1.
- Roles of the two neural networks and learned quantities: Different parts of the manuscript appear to attribute different parameter sets to the regionalization network. Please make explicit, in one place, which parameters each network predicts or corrects, their units and ranges, and how parameter scaling/normalization is handled to avoid ill-conditioning due to heterogeneous magnitudes.
- Notation and naming: Since the two networks serve distinct purposes, consider replacing generic labels (φ1, φ2) with intuitive names.
- Numerical solver choices and stability: You motivate the implicit ODE solver, which in turn needs Jacobians calculations. Please compare against an explicit solver (e.g., RK methods) in terms of accuracy, stability, computational cost, and training convergence, at least on a representative subset. Also, please report typical Newton–Raphson iteration counts, convergence criteria, and any damping/line-search strategies used to ensure robustness.
- PDE and differentiability: If the routing/finite-difference step (Eq. 7) is part of the training graph, please clarify whether gradients are backpropagated through the routing solver in addition to the ODE solver, and outline how this is implemented.
- Definition of "neutralized'' inputs: Please define precisely what is meant by "neutralized" inputs and where this is applied in the pipeline.
- Closure and derivations: Provide a short derivation or a clear reference for the closure relation in Eq. (6).
- Model capacity and regularization: Specify the architecture sizes (layers, hidden units, activations) for both networks, parameter counts, and any regularization used.
- Training strategy: During the pre-calibration phase, have you considered training the regionalization network with the original conceptual model (i.e., without the flux-correction)? Also, how crucial was the pre-calibration to the overall training performance?
Citation: https://doi.org/10.5194/egusphere-2025-2797-RC1 -
RC2: 'Comment on egusphere-2025-2797', Anonymous Referee #2, 24 Aug 2025
Review:
The authors embed a small neural network inside a physically based, gridded rainfall–runoff model and solve the resulting neural ODEs using an implicit Euler/Newton–Raphson scheme. They also learn spatially varying parameters from physical descriptors via MLPs/CNNs. In the Aude basin (France), hybrid models generally calibrate better than classical GR4 variants, and the neural-ODE approach moderates extreme runoff more plausibly during floods.
This research highlights the significance of physics-based deep learning, specifically developing a neural network to estimate fluxes and localized parameters in ODEs. It is innovative enough to be relevant to this journey. In the AI for science field, physics-guided AI is becoming increasingly important because it can be more interpretable and reliable. Additionally, it can significantly enhance the performance of traditional models based solely on physics rules.
To be honest, I am not an expert in the field of river runoff, although I have some knowledge of hybrid modeling. Therefore, for readers like myself, despite an understanding of the overall methodology, I still find it challenging to fully grasp your methods. Additionally, I would find it difficult to accept that calibrating and validating your advanced model solely within limited areas of the Aude Basin is sufficient. It would be preferable to present results from a different location to demonstrate the model's generalizability, unless one-area testing is a standard procedure in runoff modeling. Consequently, I recommend a major revision prior to the acceptance of this paper. The authors should enhance the clarity of their wording, improve the presentation of results, and include additional validation and calibration tests.
Major comments:
- As a paper on hybrid modeling, the authors should first present the general problem with a clear governing equation. Then, explicitly show what the neural networks are doing, using clear subscripts for terms predicted by NNs, such as “QNN1” and “QNN2” in this study. Next, clearly demonstrate how the NN-predicted terms are used in the governing equations. Afterwards, show the optimization process and iterations. The authors may reorganize section 2.1 to focus their system on equations (3) and (5), removing the redundant ones, and can follow examples like Brenowitz and Bretherton (2019), Yuval and O’Gorman (2020), and Yuval et al. (2021). The goal is to make this section easier for readers to understand quickly.
- From lines 84 to 91, what are the differences between GR4 and ODE? In their first appearance, it seems like GR4 functions like the host model. But later, it is all about NODE. However, the author also replaces many processes and parameterizations in GR with neural networks, resulting in GR.MLP and GR.CNN. The authors should be more straightforward about why they introduce both GR4 and ODE simultaneously—are they trying to show that one is better? What implications does that have? The authors should also explain GR4 and ODE more from a physics perspective at the beginning, clearly stating their purposes rather than just referencing them. It would be helpful to include a diagram to illustrate the workflow of GR.NN, NODE.NN, and their variants.
- Include at least one subsection about neural networks, such as MLPs or CNNs. Most importantly, I still do not know what the input variables are for both NNs. Even though they are not complex neural networks, please write about their basic architecture and hyperparameters. I know some information is already in section 2.3. Please refine it and make it easier to see, such as by adding a small table, rather than hiding it within lines.
- Although this paper focuses on hybrid models, comparing them to a pure-ML baseline would be beneficial. If it requires too much work, including references to give readers a concept of the accuracy of pure-ML models in simulating river runoffs would also be valuable.
- The authors have shown the horizontal resolution is 500m or 1km, and a time step of 1 hour. Could they also state how many grid cells are in the region for calibration? Also how much of the GPU/CPU time are used for GRs and NODEs? Will adding the neural net components significantly add to the computational burden of the host model? In Newton iterations, please show the convergence tolerances and the usual iteration steps.
- It is better to add multi-basin tests (at least one contrasting basin) to demonstrate the generalization capability and robustness of the NN parameters and the NODE system.
- Are the physical budgets constrained? For example, water conservation. In the runoff scenario, it would be storage = rainfall – ET – runoff. So, plot the cumulative rainfall – ET – storage – runoff closure and show how the NNs affect them.
Miner comments:
- Line14: Check the font style of "smash." Should it be capitalized or enclosed in quotation marks?
- Line 116: What does “neutralized” mean here for precipitation and evaporation?
- Line 120: “The neural network ϕ takes the model states as part of its inputs, thus affecting the model dynamics and state gradient information. It is expected to learn the model behavior by leveraging memory effects through state updates.” I do not know how the memory effects is learned by the neural network? Please explain.
- Figure 2. Please use the specific field names in the caption instead of
- Figure 6. Please flip the histogram of precipitation. An inverted view is difficult to interpret. Additionally, it would be better to combine the left three panels into one, increase their heights, and do the same for the right panels. This will make it easier to compare different models.
- Section 4. This section is not appropriate for this paper, which is neither a review nor an opinion piece on physics-guided AI. It should be condensed into a paragraph and added to the conclusion section.
Citation: https://doi.org/10.5194/egusphere-2025-2797-RC2 - AC2: 'Authors’ Response to Reviewers EGUSPHERE-2025-2797', Ngo Nghi Truyen Huynh, 05 Sep 2025
Interactive discussion
Status: closed
-
CEC1: 'Comment on egusphere-2025-2797', Juan Antonio Añel, 25 Jul 2025
Dear authors,
After checking your manuscript I miss the output files produced during your work. The code and input data used are correctly stored. Therefore, please, publish the output data of your work in a repository acceptable according to the Data policy of our journal, and reply to this comment with its link and permanent handler (for example a DOI). Also, remember to include in any future version of your manuscript such information for the new repository.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2025-2797-CEC1 -
AC1: 'Reply on CEC1', Ngo Nghi Truyen Huynh, 25 Jul 2025
Dear Dr. Juan A. Añel,
Thank you very much for your comment regarding the output data. Please find below the Zenodo deposit containing the output results and the scripts used to perform our study with SMASH:
https://doi.org/10.5281/zenodo.16419642This deposit includes scripts to perform calibration/validation and the output files containing the calibrated models/parameters, simulated responses, and metric scores for all 9 models compared in our paper. A detailed description of the contents is provided in the README file of the deposit.
And yes we will include in any future version of our manuscript this new repository in the data and code availability section.
Thank you again for posting your comment.
Best regards,
Ngo Nghi Truyen Huynh, on behalf of the authors
Citation: https://doi.org/10.5194/egusphere-2025-2797-AC1 -
CEC2: 'Reply on AC1', Juan Antonio Añel, 28 Jul 2025
Dear authors,
Many thanks for addressing the mentioned issue so quickly. We can consider the current version of your manuscript in compliance with the Code and Data Policy of the journal.
Juan A. Añel
Geosci. Model Dev. Executive Editor
Citation: https://doi.org/10.5194/egusphere-2025-2797-CEC2
-
CEC2: 'Reply on AC1', Juan Antonio Añel, 28 Jul 2025
-
AC1: 'Reply on CEC1', Ngo Nghi Truyen Huynh, 25 Jul 2025
-
RC1: 'Comment on egusphere-2025-2797', Anonymous Referee #1, 23 Aug 2025
Summary
The manuscript proposes a hybrid physics–AI framework for spatially distributed rainfall–runoff modeling. It builds on a differentiable, continuous state-space GR-type hydrological core with 1D routing based on a D8 drainage scheme, augmented by two neural components: (i) a regionalization network that maps physical descriptors to spatially varying hydrological parameters, and (ii) a flux-correction network that adjusts internal model fluxes using meteorological inputs and prior model states. Because state evolution is governed by ODEs, the flux-correction component is trained jointly with an implicit ODE solver in a neural-ODE fashion. Across case studies, the hybrid configurations deliver more accurate and stable streamflow predictions than the classical GR baseline. Overall, this is a timely and valuable contribution to robust, differentiable coupling of process-based hydrology with machine learning.
Major comments- Self-contained presentation and background: The current manuscript would benefit from a concise background on rainfall–runoff modeling (empirical/black-box, conceptual, and distributed approaches; recent ML-based advances and their limitations) to situate the contribution and clarify what is new versus prior work.
- High-level system overview: Given the number of interacting components (regionalization network, flux-correction network, ODE state evolution, PDE/routing), please add a clear schematic that shows data flow for the different configurations listed in Table 1.
- Roles of the two neural networks and learned quantities: Different parts of the manuscript appear to attribute different parameter sets to the regionalization network. Please make explicit, in one place, which parameters each network predicts or corrects, their units and ranges, and how parameter scaling/normalization is handled to avoid ill-conditioning due to heterogeneous magnitudes.
- Notation and naming: Since the two networks serve distinct purposes, consider replacing generic labels (φ1, φ2) with intuitive names.
- Numerical solver choices and stability: You motivate the implicit ODE solver, which in turn needs Jacobians calculations. Please compare against an explicit solver (e.g., RK methods) in terms of accuracy, stability, computational cost, and training convergence, at least on a representative subset. Also, please report typical Newton–Raphson iteration counts, convergence criteria, and any damping/line-search strategies used to ensure robustness.
- PDE and differentiability: If the routing/finite-difference step (Eq. 7) is part of the training graph, please clarify whether gradients are backpropagated through the routing solver in addition to the ODE solver, and outline how this is implemented.
- Definition of "neutralized'' inputs: Please define precisely what is meant by "neutralized" inputs and where this is applied in the pipeline.
- Closure and derivations: Provide a short derivation or a clear reference for the closure relation in Eq. (6).
- Model capacity and regularization: Specify the architecture sizes (layers, hidden units, activations) for both networks, parameter counts, and any regularization used.
- Training strategy: During the pre-calibration phase, have you considered training the regionalization network with the original conceptual model (i.e., without the flux-correction)? Also, how crucial was the pre-calibration to the overall training performance?
Citation: https://doi.org/10.5194/egusphere-2025-2797-RC1 -
RC2: 'Comment on egusphere-2025-2797', Anonymous Referee #2, 24 Aug 2025
Review:
The authors embed a small neural network inside a physically based, gridded rainfall–runoff model and solve the resulting neural ODEs using an implicit Euler/Newton–Raphson scheme. They also learn spatially varying parameters from physical descriptors via MLPs/CNNs. In the Aude basin (France), hybrid models generally calibrate better than classical GR4 variants, and the neural-ODE approach moderates extreme runoff more plausibly during floods.
This research highlights the significance of physics-based deep learning, specifically developing a neural network to estimate fluxes and localized parameters in ODEs. It is innovative enough to be relevant to this journey. In the AI for science field, physics-guided AI is becoming increasingly important because it can be more interpretable and reliable. Additionally, it can significantly enhance the performance of traditional models based solely on physics rules.
To be honest, I am not an expert in the field of river runoff, although I have some knowledge of hybrid modeling. Therefore, for readers like myself, despite an understanding of the overall methodology, I still find it challenging to fully grasp your methods. Additionally, I would find it difficult to accept that calibrating and validating your advanced model solely within limited areas of the Aude Basin is sufficient. It would be preferable to present results from a different location to demonstrate the model's generalizability, unless one-area testing is a standard procedure in runoff modeling. Consequently, I recommend a major revision prior to the acceptance of this paper. The authors should enhance the clarity of their wording, improve the presentation of results, and include additional validation and calibration tests.
Major comments:
- As a paper on hybrid modeling, the authors should first present the general problem with a clear governing equation. Then, explicitly show what the neural networks are doing, using clear subscripts for terms predicted by NNs, such as “QNN1” and “QNN2” in this study. Next, clearly demonstrate how the NN-predicted terms are used in the governing equations. Afterwards, show the optimization process and iterations. The authors may reorganize section 2.1 to focus their system on equations (3) and (5), removing the redundant ones, and can follow examples like Brenowitz and Bretherton (2019), Yuval and O’Gorman (2020), and Yuval et al. (2021). The goal is to make this section easier for readers to understand quickly.
- From lines 84 to 91, what are the differences between GR4 and ODE? In their first appearance, it seems like GR4 functions like the host model. But later, it is all about NODE. However, the author also replaces many processes and parameterizations in GR with neural networks, resulting in GR.MLP and GR.CNN. The authors should be more straightforward about why they introduce both GR4 and ODE simultaneously—are they trying to show that one is better? What implications does that have? The authors should also explain GR4 and ODE more from a physics perspective at the beginning, clearly stating their purposes rather than just referencing them. It would be helpful to include a diagram to illustrate the workflow of GR.NN, NODE.NN, and their variants.
- Include at least one subsection about neural networks, such as MLPs or CNNs. Most importantly, I still do not know what the input variables are for both NNs. Even though they are not complex neural networks, please write about their basic architecture and hyperparameters. I know some information is already in section 2.3. Please refine it and make it easier to see, such as by adding a small table, rather than hiding it within lines.
- Although this paper focuses on hybrid models, comparing them to a pure-ML baseline would be beneficial. If it requires too much work, including references to give readers a concept of the accuracy of pure-ML models in simulating river runoffs would also be valuable.
- The authors have shown the horizontal resolution is 500m or 1km, and a time step of 1 hour. Could they also state how many grid cells are in the region for calibration? Also how much of the GPU/CPU time are used for GRs and NODEs? Will adding the neural net components significantly add to the computational burden of the host model? In Newton iterations, please show the convergence tolerances and the usual iteration steps.
- It is better to add multi-basin tests (at least one contrasting basin) to demonstrate the generalization capability and robustness of the NN parameters and the NODE system.
- Are the physical budgets constrained? For example, water conservation. In the runoff scenario, it would be storage = rainfall – ET – runoff. So, plot the cumulative rainfall – ET – storage – runoff closure and show how the NNs affect them.
Miner comments:
- Line14: Check the font style of "smash." Should it be capitalized or enclosed in quotation marks?
- Line 116: What does “neutralized” mean here for precipitation and evaporation?
- Line 120: “The neural network ϕ takes the model states as part of its inputs, thus affecting the model dynamics and state gradient information. It is expected to learn the model behavior by leveraging memory effects through state updates.” I do not know how the memory effects is learned by the neural network? Please explain.
- Figure 2. Please use the specific field names in the caption instead of
- Figure 6. Please flip the histogram of precipitation. An inverted view is difficult to interpret. Additionally, it would be better to combine the left three panels into one, increase their heights, and do the same for the right panels. This will make it easier to compare different models.
- Section 4. This section is not appropriate for this paper, which is neither a review nor an opinion piece on physics-guided AI. It should be condensed into a paragraph and added to the conclusion section.
Citation: https://doi.org/10.5194/egusphere-2025-2797-RC2 - AC2: 'Authors’ Response to Reviewers EGUSPHERE-2025-2797', Ngo Nghi Truyen Huynh, 05 Sep 2025
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Ngo Nghi Truyen Huynh
Pierre-André Garambois
François Colleoni
Jérôme Monnier
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(3055 KB) - Metadata XML
Dear authors,
After checking your manuscript I miss the output files produced during your work. The code and input data used are correctly stored. Therefore, please, publish the output data of your work in a repository acceptable according to the Data policy of our journal, and reply to this comment with its link and permanent handler (for example a DOI). Also, remember to include in any future version of your manuscript such information for the new repository.
Juan A. Añel
Geosci. Model Dev. Executive Editor