Huynh et al. (2026) A hybrid physics–AI approach using universal differential equations with state-dependent neural networks for learnable, regionalizable, spatially distributed hydrological modeling

Identification

• Journal: Geoscientific model development
• Year: 2026
• Date: 2026-02-02
• Authors: Ngo Nghi Truyen Huynh, Pierre-André Garambois, François Colleoni, Jérôme Monnier
• DOI: 10.5194/gmd-19-1055-2026

Research Groups

• INRAE RECOVER, Aix-Marseille Université, Aix-en-Provence, France
• INSA, Institut de Mathématiques de Toulouse (IMT), Université de Toulouse, Toulouse, France

Short Summary

This study introduces a hybrid physics–AI framework that integrates state-dependent neural networks into a spatially distributed, regionalizable, and fully differentiable hydrological model using universal differential equations (UDEs). The framework demonstrates consistently strong performance in streamflow simulations, particularly for flood modeling, by refining internal water fluxes and improving parameter regionalization.

Objective

• To propose a hybrid physics–AI approach that integrates state-dependent neural networks within universal differential equations (UDEs) into a differentiable, regionalizable, and spatially distributed hydrological model, solved using an implicit numerical scheme.
• To enhance flood modeling, mitigate structural uncertainty, optimize data efficiency, and enable more effective multi-scale information extraction through hybrid flux correction.

Study Configuration

• Spatial Scale: Aude River basin (southern France), approximately 10 400 square kilometres (104 × 100 grid) with an active domain of 4902 square kilometres (4902 active cells), comprising 25 catchments. Models run at a spatial resolution of 1 kilometre.
• Temporal Scale: Hourly time step. Study period spans 9 years (August 2014 to July 2023), with calibration over 2015–2019 (P1) and validation over 2019–2023 (P2), including one-year warm-up periods.

Methodology and Data

• Models used:
  • Hybrid physics–AI framework based on Universal Differential Equations (UDEs) with state-dependent Neural Networks (NNs).
  • Continuous state-space GR4-like hydrological model.
  • Kinematic wave routing model (Partial Differential Equation).
  • Regionalization NNs: Multilayer Perceptrons (MLPs) and Convolutional Neural Networks (CNNs).
  • Process-parameterization NN: MLP.
  • Implicit Euler scheme with Newton–Raphson method for UDE/ODE resolution.
  • Adam optimizer for gradient-based training.
  • Tapenade engine for automatic differentiation.
  • smash platform (version 1.1).
• Data sources:
  • National database covering Metropolitan France (multi-source data).
  • Preprocessed data from SCHAPI-DGPR and Météo-France.
  • Observed discharge time series.
  • Seven physical descriptors (local slope, drainage density, percentage of basin area in karst zone, forest cover rate, urban cover rate, potential available water reserve, high storage capacity basin rate) at 0.01° resolution.
  • Digital Elevation Model (DEM) for D8 drainage scheme.

Main Results

• Hybrid approaches demonstrated consistently strong and stable performance across calibration and various validation scenarios.
• Model calibration scores increased with the complexity of regionalization mappings (from uniform to MLP and CNN). For example, UDE.CNN achieved a median Nash–Sutcliffe Efficiency (NSE) over 0.85 in calibration, compared to 0.82 for ODE.CNN and 0.8 for GR.CNN.
• In temporal validation, ODE.MLP and UDE.MLP yielded similar high median NSE scores of 0.71, outperforming GR.MLP (0.59).
• The UDE structure exhibited a hybridization effect, modifying state dynamics and runoff flow, leading to more accurate streamflow simulations for flood modeling.
• CNN-based regionalization models produced smoother parameter maps compared to MLP-based models, reflecting spatial patterns from physical descriptors.
• The UDE.MLP model showed higher values and greater variability in the transfer state compared to ODE.MLP and GR.MLP, indicating increased rainfall sensitivity due to the process-parameterization NN.
• Uniform mapping models significantly underestimated flood magnitudes, while regionalization approaches (MLP and CNN) improved flood simulation accuracy (e.g., RMSE values of 4.37 cubic metres per second for ODE.MLP and 4.13 cubic metres per second for UDE.MLP compared to 6.2 cubic metres per second for GR.MLP at an upstream gauge).

Contributions

• Introduced a novel hybrid physics–AI framework that seamlessly integrates state-dependent neural networks into a spatialized, regionalizable, and fully differentiable process-based hydrological model using Universal Differential Equations (UDEs).
• Developed a mathematically rigorous method for computing Jacobian matrices required by implicit numerical schemes when incorporating state-dependent neural network components within UDEs.
• Extended the regionalization mapping to include Convolutional Neural Networks (CNNs), enhancing the adaptability and scalability of spatially distributed parameter estimation by capturing spatial dependencies.
• Implemented and released this framework as smash v1.1, significantly expanding the platform's capabilities for comprehensive evaluation of hybrid hydrological models at kilometric spatial and hourly temporal resolutions.
• Demonstrated that the UDE structure, by refining internal water fluxes, improves the representation of hydrological responses and achieves more accurate streamflow simulations, particularly for flood events.

Funding

• ANR MUFFINS project (MUltiscale Flood Forecasting with INnovating Solutions), grant no. ANR-21-CE04-0021-01.
• NEPTUNE European project DG-ECO.

Citation

@article{Huynh2026hybrid,
  author = {Huynh, Ngo Nghi Truyen and Garambois, Pierre-André and Colleoni, François and Monnier, Jérôme},
  title = {A hybrid physics–AI approach using universal differential equations with state-dependent neural networks for learnable, regionalizable, spatially distributed hydrological modeling},
  journal = {Geoscientific model development},
  year = {2026},
  doi = {10.5194/gmd-19-1055-2026},
  url = {https://doi.org/10.5194/gmd-19-1055-2026}
}