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Cited by 38Hydrological Forecasting Using AIHydrology and Watershed Management StudiesFlood Risk Assessment and Management

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A distributed hybrid physics–AI framework for learning corrections of internal hydrological fluxes and enhancing high-resolution regionalized flood modeling

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Huynh, Ngo Nghi Truyen, Garambois, Pierre-André, Renard, Benjamin, Colleoni, François, Monnier, Jérôme, Roux, Hélène·2025·Hydrology and Earth System Sciences
DOI 10.5194/hess-29-3589-2025

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Abstract

To advance the discovery of scale-relevant hydrological laws while better exploiting massive multisource data, merging artificial intelligence with process-based modeling has emerged as a compelling approach, as demonstrated in recent lumped hydrological modeling studies. This research proposes a general spatially distributed hybrid modeling framework that seamlessly combines differentiable process-based models with neural networks. Specifically, we focus on hybridizing the hydrological module – built atop a differentiable kinematic wave routing over a flow direction grid – using a process-parameterization network that refines internal water fluxes, with all conceptual parameters estimated by a regionalization network trained simultaneously. We evaluate flood modeling performance and analyze the interpretability of learned conceptual parameters and corrections of internal fluxes using two high-resolution datasets (d x =1 km and d t =1 h). The first dataset involves 235 catchments in France, used for local calibration–validation and model structure comparisons between the classical Génie Rural (GR)-like model and the hybrid approach. The second dataset presents a challenging multi-catchment modeling setup in flash-flood-prone areas to demonstrate the framework's regionalization learning capabilities. The results show that the hybrid models achieve superior accuracy and robustness compared to classical approaches in both spatial and temporal validation. Analysis of the spatially distributed parameters and internal fluxes reveals the hybrid models' nuanced behavior, their adaptability to diverse hydrological responses, and their potential to uncover physical processes.

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Figure 1 . Figure 1. Hybrid physics-AI framework, applied to the spatially distributed GR-like and kinematic wave model, involving a pair of processparameterization and regionalization neural networks. The pair of neural networks is used to (i) correct internal fluxes (using neutralized atmospheric data) and (ii) estimate the model parameters (using physical descriptors), with their weights optimized through high-dimensional optimization algorithms using an adjoint model to obtain accurate gradients of the cost function.
Figure 1 . Figure 1. Hybrid physics-AI framework, applied to the spatially distributed GR-like and kinematic wave model, involving a pair of processparameterization and regionalization neural networks. The pair of neural networks is used to (i) correct internal fluxes (using neutralized atmospheric data) and (ii) estimate the model parameters (using physical descriptors), with their weights optimized through high-dimensional optimization algorithms using an adjoint model to obtain accurate gradients of the cost function.CC-BY · From the PDF
Figure 2 . Figure 2. Study areas used for evaluation. The first area consists of 235 non-nested catchments in France, while the second area includes 21 catchments in a multiple-catchment setup in the Eastern Mediterranean region, representing contrasting hydrological conditions.
Figure 2 . Figure 2. Study areas used for evaluation. The first area consists of 235 non-nested catchments in France, while the second area includes 21 catchments in a multiple-catchment setup in the Eastern Mediterranean region, representing contrasting hydrological conditions.CC-BY · From the PDF
Figure 3 . Figure 3. Comparison of streamflow simulation across representative small, medium, and large catchments randomly selected among the 235 catchments in France during the validation period.
Figure 3 . Figure 3. Comparison of streamflow simulation across representative small, medium, and large catchments randomly selected among the 235 catchments in France during the validation period.CC-BY · From the PDF
Figure 4 . Figure 4. Model performance comparison of local calibration methods across 235 catchments in France. The two boxes in lighter colors represent the classical GR models (GR.U and GR.D), while the two boxes in darker colors represent the ϕ1-hybrid models (GRNN.U and GRNN.D). The evaluation is based on simulated discharges over the calibration and validation period, using NSE, KGE, and RMSE metrics.
Figure 4 . Figure 4. Model performance comparison of local calibration methods across 235 catchments in France. The two boxes in lighter colors represent the classical GR models (GR.U and GR.D), while the two boxes in darker colors represent the ϕ1-hybrid models (GRNN.U and GRNN.D). The evaluation is based on simulated discharges over the calibration and validation period, using NSE, KGE, and RMSE metrics.CC-BY · From the PDF
Figure 5 . Figure 5. Comparison of model performance in simulating flood event signatures, presented as the cumulative distribution function (CDF) of the relative error (RE) between observed and simulated values for peak flow (Epf), runoff coefficient (Erc), flood flow (Eff), and baseflow (Ebf). The evaluation is based on 2,718 flood events across 235 catchments during the validation period (08/2013-07/2019).
Figure 5 . Figure 5. Comparison of model performance in simulating flood event signatures, presented as the cumulative distribution function (CDF) of the relative error (RE) between observed and simulated values for peak flow (Epf), runoff coefficient (Erc), flood flow (Eff), and baseflow (Ebf). The evaluation is based on 2,718 flood events across 235 catchments during the validation period (08/2013-07/2019).CC-BY · From the PDF
Figure 6 . Figure 6. Comparison of multi-catchment regionalization performance for different methods. The evaluation is based on NSE scores computed over three periods: the 3-year calibration period (excluding a 1-year warmup), the first 18 months, and the last 18 months of the 3-year validation period for 11 calibration catchments and 10 validation catchments. The numbers in parentheses on the boxplots indicate the total number of samples evaluated, with the validation period split into two 18-month periods (sample counts are doubled for all catchments).
Figure 6 . Figure 6. Comparison of multi-catchment regionalization performance for different methods. The evaluation is based on NSE scores computed over three periods: the 3-year calibration period (excluding a 1-year warmup), the first 18 months, and the last 18 months of the 3-year validation period for 11 calibration catchments and 10 validation catchments. The numbers in parentheses on the boxplots indicate the total number of samples evaluated, with the validation period split into two 18-month periods (sample counts are doubled for all catchments).CC-BY · From the PDF
Figure 7 . Figure 7. (a) Comparison of NSE and RMSE metrics computed for 143 flood events detected in the validation period across 21 catchments in the MedEst region. (b) Comparison of streamflow simulations for representative small, medium, and large catchments during several flood events selected in the validation period.
Figure 7 . Figure 7. (a) Comparison of NSE and RMSE metrics computed for 143 flood events detected in the validation period across 21 catchments in the MedEst region. (b) Comparison of streamflow simulations for representative small, medium, and large catchments during several flood events selected in the validation period.CC-BY · From the PDF
Figure 8 . Figure 8. Maps of spatio-temporal average flux corrections g f q x∈Ω j t , where j = 1..Ng, for the Ng = 235 catchments, obtained through local calibrations of spatially uniform parameters with the hybrid model structure (GRNN.U). The function g(.) represents the transformation applied to the neural network output f q , which may differ depending on the specific flux being corrected. Red indicates corrections that tend to increase the current flux, while blue indicates corrections that reduce it, with white representing minimal or no effective correction.
Figure 8 . Figure 8. Maps of spatio-temporal average flux corrections g f q x∈Ω j t , where j = 1..Ng, for the Ng = 235 catchments, obtained through local calibrations of spatially uniform parameters with the hybrid model structure (GRNN.U). The function g(.) represents the transformation applied to the neural network output f q , which may differ depending on the specific flux being corrected. Red indicates corrections that tend to increase the current flux, while blue indicates corrections that reduce it, with white representing minimal or no effective correction.CC-BY · From the PDF
Figure 9 . Figure 9. (a) Heatmap of spatially averaged flux correction time series g f q (t) x∈Ω j , where j = 1..Ng, for the Ng = 235 catchments. These are obtained through local calibrations of spatially uniform parameters with the hybrid model structure (GRNN.U). (b) Boxplot of spatio-temporal average flux corrections g f q x∈Ω j t, where each boxplot represents 235 catchment-specific spatio-temporal averages.
Figure 9 . Figure 9. (a) Heatmap of spatially averaged flux correction time series g f q (t) x∈Ω j , where j = 1..Ng, for the Ng = 235 catchments. These are obtained through local calibrations of spatially uniform parameters with the hybrid model structure (GRNN.U). (b) Boxplot of spatio-temporal average flux corrections g f q x∈Ω j t, where each boxplot represents 235 catchment-specific spatio-temporal averages.CC-BY · From the PDF
Figure 10 . Figure 10. Visualization of flux corrections in the MedEst region obtained through regional calibration of spatially uniform parameters with the hybrid model (GRNN.U): (a) Spatial average of infiltrating flux correction 1 + fq,1(t) x , original and corrected infiltrating rainfall Ps(t) x , Ps(t) x ; (b) Maps of time-averaged infiltrating flux correction 1 + fq,1(x) t , original and corrected infiltrating rainfall Ps(x) t , Ps(x) t , where µ and σ represent the spatial average and standard deviation.
Figure 10 . Figure 10. Visualization of flux corrections in the MedEst region obtained through regional calibration of spatially uniform parameters with the hybrid model (GRNN.U): (a) Spatial average of infiltrating flux correction 1 + fq,1(t) x , original and corrected infiltrating rainfall Ps(t) x , Ps(t) x ; (b) Maps of time-averaged infiltrating flux correction 1 + fq,1(x) t , original and corrected infiltrating rainfall Ps(x) t , Ps(x) t , where µ and σ represent the spatial average and standard deviation.CC-BY · From the PDF
Figure 11 . Figure 11. Comparison of mass fluxes affecting water balance in interannual mean at basin scale over the 235 catchment set, for the classical GR model (GR.U) and the ϕ1-hybrid model (GRNN.U) using uniform mapping "U" for conceptual parameters θ, in local calibration. The x-axis shows 235 catchments, sorted by their average precipitation. Grey dots represent the interannual averages of precipitation P , actual evapotranspiration Es, exchange flux F , and lateral discharge Q lat over the calibration period for the classical model. Red dots and lines represent increases according to the hybrid model, while blue ones indicate decreases. For cases where F < 0, red indicates a larger magnitude in F for the hybrid model (more negative), while blue indicates a lesser magnitude (closer to zero).
Figure 11 . Figure 11. Comparison of mass fluxes affecting water balance in interannual mean at basin scale over the 235 catchment set, for the classical GR model (GR.U) and the ϕ1-hybrid model (GRNN.U) using uniform mapping "U" for conceptual parameters θ, in local calibration. The x-axis shows 235 catchments, sorted by their average precipitation. Grey dots represent the interannual averages of precipitation P , actual evapotranspiration Es, exchange flux F , and lateral discharge Q lat over the calibration period for the classical model. Red dots and lines represent increases according to the hybrid model, while blue ones indicate decreases. For cases where F < 0, red indicates a larger magnitude in F for the hybrid model (more negative), while blue indicates a lesser magnitude (closer to zero).CC-BY · From the PDF
Figure 12 . Figure 12. Hydrological interpretation of the non-linear response surface obtained using the learned flux correction neural network for infiltrating rainfall Ps = (1+fq,1)Ps, plotted with a production capacity of cp = 15 mm and neutralized evapotranspiration of En = 0 mm: (a) Original and corrected net rainfall Pr for different levels of the production state hp; (b) Response surface of original and corrected net rainfall Pr as a function of both the production state hp and neutralized rainfall Pn.
Figure 12 . Figure 12. Hydrological interpretation of the non-linear response surface obtained using the learned flux correction neural network for infiltrating rainfall Ps = (1+fq,1)Ps, plotted with a production capacity of cp = 15 mm and neutralized evapotranspiration of En = 0 mm: (a) Original and corrected net rainfall Pr for different levels of the production state hp; (b) Response surface of original and corrected net rainfall Pr as a function of both the production state hp and neutralized rainfall Pn.CC-BY · From the PDF
Figure B1 . Figure B1. Maps of spatio-temporal average flux corrections g f q x∈Ω j t , where j = 1..Ng, for the Ng = 235 catchments, obtained through local calibrations of spatially distributed parameters with the hybrid model structure (GRNN.D). The function g(.) represents the transformation applied to the neural network output f q , which may differ depending on the specific flux being corrected. Red indicates corrections that tend to increase the current flux, while blue indicates corrections that reduce it, with white representing minimal or no effective correction.
Figure B1 . Figure B1. Maps of spatio-temporal average flux corrections g f q x∈Ω j t , where j = 1..Ng, for the Ng = 235 catchments, obtained through local calibrations of spatially distributed parameters with the hybrid model structure (GRNN.D). The function g(.) represents the transformation applied to the neural network output f q , which may differ depending on the specific flux being corrected. Red indicates corrections that tend to increase the current flux, while blue indicates corrections that reduce it, with white representing minimal or no effective correction.CC-BY · From the PDF
Figure B2 . Figure B2. (a) Heatmap of spatially averaged flux correction time series g f q (t) x∈Ω j , where j = 1, . . . , Ng, for the Ng = 235 catchments. These are obtained through local calibrations of spatially distributed parameters with the hybrid model structure (GRNN.D). (b) Boxplot of spatio-temporal average flux corrections g f q x∈Ω j t, where each boxplot represents 235 catchment-specific spatio-temporal averages.
Figure B2 . Figure B2. (a) Heatmap of spatially averaged flux correction time series g f q (t) x∈Ω j , where j = 1, . . . , Ng, for the Ng = 235 catchments. These are obtained through local calibrations of spatially distributed parameters with the hybrid model structure (GRNN.D). (b) Boxplot of spatio-temporal average flux corrections g f q x∈Ω j t, where each boxplot represents 235 catchment-specific spatio-temporal averages.CC-BY · From the PDF
Figure B3 . Figure B3. Comparison of mass fluxes affecting water balance in interannual mean at basin scale over the 235 catchment set, for the classical GR model (GR.D) and the ϕ1-hybrid model (GRNN.D) using distributed mapping "D" for conceptual parameters θ, in local calibration. The x-axis shows 235 catchments, sorted by their average precipitation. Grey dots represent the interannual averages of precipitation P , actual evapotranspiration Es, exchange flux F , and lateral discharge Q lat over the calibration period for the classical model. Red dots and lines represent increases according to the hybrid model, while blue ones indicate decreases. For cases where F < 0, red indicates a larger magnitude in F for the hybrid model (more negative), while blue indicates a lesser magnitude (closer to zero).
Figure B3 . Figure B3. Comparison of mass fluxes affecting water balance in interannual mean at basin scale over the 235 catchment set, for the classical GR model (GR.D) and the ϕ1-hybrid model (GRNN.D) using distributed mapping "D" for conceptual parameters θ, in local calibration. The x-axis shows 235 catchments, sorted by their average precipitation. Grey dots represent the interannual averages of precipitation P , actual evapotranspiration Es, exchange flux F , and lateral discharge Q lat over the calibration period for the classical model. Red dots and lines represent increases according to the hybrid model, while blue ones indicate decreases. For cases where F < 0, red indicates a larger magnitude in F for the hybrid model (more negative), while blue indicates a lesser magnitude (closer to zero).CC-BY · From the PDF
Figure B4 . Figure B4. Comparison of mean absolute error (MAE), percent bias (PBIAS), and peak flow ratio (PFR) metrics computed for 143 flood events detected in the validation period across 21 catchments in the MedEst region.
Figure B4 . Figure B4. Comparison of mean absolute error (MAE), percent bias (PBIAS), and peak flow ratio (PFR) metrics computed for 143 flood events detected in the validation period across 21 catchments in the MedEst region.CC-BY · From the PDF
Figure B5 . Figure B5. Visualization of flux corrections in the MedEst region obtained through regional calibration of spatially distributed parameters with the fully integrated hybrid model (GRNN.NN): (a) Spatial average of infiltrating flux correction 1 + fq,1(t)x , original and corrected infiltrating rainfall Ps(t)x , Ps(t)x ; (b) Maps of time-averaged infiltrating flux correction 1 + fq,1(x) t , original and corrected infiltrating rainfall Ps(x) t , Ps(x) t , where µ and σ represent the spatial average and standard deviation.
Figure B5 . Figure B5. Visualization of flux corrections in the MedEst region obtained through regional calibration of spatially distributed parameters with the fully integrated hybrid model (GRNN.NN): (a) Spatial average of infiltrating flux correction 1 + fq,1(t)x , original and corrected infiltrating rainfall Ps(t)x , Ps(t)x ; (b) Maps of time-averaged infiltrating flux correction 1 + fq,1(x) t , original and corrected infiltrating rainfall Ps(x) t , Ps(x) t , where µ and σ represent the spatial average and standard deviation.CC-BY · From the PDF

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Metadata

Authors
Huynh, N. N. T., Garambois, P.-A., Renard, B., Colleoni, F., Monnier, J., & Roux, H.
Affiliations
  • Aix-Marseille Université (FR)
  • Institut National des Sciences Appliquées de Toulouse (FR)
  • Centre National de la Recherche Scientifique (FR)
ORCID
Venue
Hydrology and Earth System Sciences
Publisher
Copernicus GmbH
Published
2025-08-07
Volume
29
Issue
15
Pages
3589-3613
ISSN
  • 1027-5606 (ISSN-L)
  • 1607-7938
DOI
10.5194/hess-29-3589-2025
OpenAlex
W4413266363
Cited by
38
References
3
Citations by year
2026: 27 · 2025: 11
FWCI
9.45
Type
journal article
Language
en
Open access
Licence
Locations
  • Hydrology and earth system sciences (publishedVersion, cc-by)
  • HAL (submittedVersion, cc-by)
  • HAL (submittedVersion, other-oa)
  • DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Funders
Agence Nationale de la Recherche (ANR-21-CE04-0021-01)
Topics
  • Physical Sciences › Environmental Science › Environmental Engineering › Hydrological Forecasting Using AI
  • Physical Sciences › Environmental Science › Water Science and Technology › Hydrology and Watershed Management Studies
  • Physical Sciences › Environmental Science › Global and Planetary Change › Flood Risk Assessment and Management
Keywords
distributed hydrological model, kinematic wave routing

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