A library of hydrological models developed on PyTorch and designed alongside 𝛿MG for the creation of end-to-end differentiable models, enabling parameter learning, bias correction, missing process representation, and more.

See 𝛿MG/examples using HydroDL2-based HBV models for published differentiable parameter learning (dPL) applications, and see citation for details on individual model architectures.

uv pip install hydrodl2

For development installs, see setup.

import hydrodl2

# List all available models
hydrodl2.available_models()
# {'hbv': ['hbv', 'hbv_1_1p', 'hbv_2', 'hbv_2_hourly', 'hbv_2_mts', 'hbv_adj']}

# Load a model class
Hbv = hydrodl2.load_model('hbv')

# Instantiate and use in a differentiable pipeline
model = Hbv()

Models are standard torch.nn.Module subclasses and can be composed with neural networks via δMG for end-to-end differentiable training.

.
├── src/
│   └── hydrodl2/
│       ├── api/                   # Main API
│       │   ├── __init__.py
│       │   └── methods.py         # Methods exposed to end-users
│       ├── core/                  # Methods used internally
│       ├── models/                # Shared models directory
│       │   └── hbv/               # HBV model variants
│       └── modules/               # Augmentations for δMG models
├── docs/
├── tests/
└── pyproject.toml

This work is maintained by MHPI and advised by Dr. Chaopeng Shen. If you find it useful, please cite:

Shen, C., Appling, A.P., Gentine, P. et al. Differentiable modelling to unify machine learning and physical models for geosciences. Nat Rev Earth Environ 4, 552–567 (2023). https://doi.org/10.1038/s43017-023-00450-9

@article{shen_differentiable_2023,
    title = {Differentiable modelling to unify machine learning and physical models for geosciences},
    volume = {4},
    issn = {2662-138X},
    url = {https://doi.org/10.1038/s43017-023-00450-9},
    doi = {10.1038/s43017-023-00450-9},
    pages = {552--567},
    number = {8},
    journaltitle = {Nature Reviews Earth \& Environment},
    author = {Shen, Chaopeng and Appling, Alison P. and Gentine, Pierre and Bandai, Toshiyuki and Gupta, Hoshin and Tartakovsky, Alexandre and Baity-Jesi, Marco and Fenicia, Fabrizio and Kifer, Daniel and Li, Li and Liu, Xiaofeng and Ren, Wei and Zheng, Yi and Harman, Ciaran J. and Clark, Martyn and Farthing, Matthew and Feng, Dapeng and Kumar, Praveen and Aboelyazeed, Doaa and Rahmani, Farshid and Song, Yalan and Beck, Hylke E. and Bindas, Tadd and Dwivedi, Dipankar and Fang, Kuai and Höge, Marvin and Rackauckas, Chris and Mohanty, Binayak and Roy, Tirthankar and Xu, Chonggang and Lawson, Kathryn},
    date = {2023-08-01},
}

Models:

1. (HBV) Feng, D., Liu, J., Lawson, K., & Shen, C. (2022). Differentiable, learnable, regionalized process-based models with multiphysical outputs can approach state-of-the-art hydrologic prediction accuracy. Water Resources Research, 58, e2022WR032404. https://doi.org/10.1029/2022WR032404

   BibTeX

   @article{https://doi.org/10.1029/2022WR032404,
       author = {Feng, Dapeng and Liu, Jiangtao and Lawson, Kathryn and Shen, Chaopeng},
       title = {Differentiable, Learnable, Regionalized Process-Based Models With Multiphysical Outputs can Approach State-Of-The-Art Hydrologic Prediction Accuracy},
       journal = {Water Resources Research},
       volume = {58},
       number = {10},
       pages = {e2022WR032404},
       keywords = {rainfall runoff, differentiable programming, machine learning, physical model, differentiable hydrology, LSTM},
       doi = {https://doi.org/10.1029/2022WR032404},
       year = {2022},
   }

   
   

2. (HBV Adj.) Song, Y., Knoben, W. J. M., Clark, M. P., Feng, D., Lawson, K., Sawadekar, K., and Shen, C.: When ancient numerical demons meet physics-informed machine learning: adjoint-based gradients for implicit differentiable modeling, Hydrol. Earth Syst. Sci., 28, 3051–3077, https://doi.org/10.5194/hess-28-3051-2024, 2024.

   BibTeX

   @Article{hess-28-3051-2024,
       AUTHOR = {Song, Y. and Knoben, W. J. M. and Clark, M. P. and Feng, D. and Lawson, K. and Sawadekar, K. and Shen, C.},
       TITLE = {When ancient numerical demons meet physics-informed machine learning:
       adjoint-based gradients for implicit differentiable modeling},
       JOURNAL = {Hydrology and Earth System Sciences},
       VOLUME = {28},
       YEAR = {2024},
       NUMBER = {13},
       PAGES = {3051--3077},
       URL = {https://hess.copernicus.org/articles/28/3051/2024/},
       DOI = {10.5194/hess-28-3051-2024}
   }

   
   

3. (HBV 1.1p) Yalan Song, Kamlesh Sawadekar, Jonathan M Frame, et al. Physics-informed, Differentiable Hydrologic Models for Capturing Unseen Extreme Events . ESS Open Archive . March 14, 2025. https://doi.org/10.22541/essoar.172304428.82707157/v2 [Accepted]

   BibTeX

   @article{https://doi.org/10.22541/essoar.172304428.82707157/v2,
       author = {Song, Yalan and Sawadekar, Kamlesh and Frame, Jonathan and Pan, Ming and Clark, Martyn and Knoben, Wouter J. M. and Wood W., Andrew and Lawson E., Kathryn and Patel, Trupesh and Shen, Chaopeng},
       title = {Physics-informed, Differentiable Hydrologic  Models for Capturing Unseen Extreme Events},
       journal = {ESS Open Archive},
       volume = {},
       number = {},
       pages = {},
       keywords = {hydrology, differentiable modeling, extremes, physics-informed machine learning, streamflow, streamflow regime},
       doi = {https://doi.org/10.22541/essoar.172304428.82707157/v2},
       year = {2025},

   
   

4. (HBV 2.0) Song, Y., Bindas, T., Shen, C., Ji, H., Knoben, W. J. M., Lonzarich, L., et al. (2025). High-resolution national-scale water modeling is enhanced by multiscale differentiable physics-informed machine learning. Water Resources Research, 61, e2024WR038928. https://doi.org/10.1029/2024WR038928

   BibTeX

   @article{https://doi.org/10.1029/2024WR038928,
       author = {Song, Yalan and Bindas, Tadd and Shen, Chaopeng and Ji, Haoyu and Knoben, Wouter J. M. and Lonzarich, Leo and Clark, Martyn P. and Liu, Jiangtao and van Werkhoven, Katie and Lamont, Sam and Denno, Matthew and Pan, Ming and Yang, Yuan and Rapp, Jeremy and Kumar, Mukesh and Rahmani, Farshid and Thébault, Cyril and Adkins, Richard and Halgren, James and Patel, Trupesh and Patel, Arpita and Sawadekar, Kamlesh Arun and Lawson, Kathryn},
       title = {High-Resolution National-Scale Water Modeling Is Enhanced by Multiscale Differentiable Physics-Informed Machine Learning},
       journal = {Water Resources Research},
       volume = {61},
       number = {4},
       pages = {e2024WR038928},
       keywords = {differentiable modeling, physics-informed machine learning, National Water Model, routing, Muskingum Cunge, multiscale training},
       doi = {https://doi.org/10.1029/2024WR038928},
       year = {2025},
   }

   
   

5. (HBV 2.0 MTS) Yang, W., Ji, H., Lonzarich, L., Song, Y., Shen, C. (2025). Diffusion-Based Probabilistic Modeling for Hourly Streamflow Prediction and Assimilation. arXiv. https://arxiv.org/abs/2510.08488 [Under Review]

   BibTeX

   @misc{yang2025diffusionbasedprobabilisticmodelinghourly,
         title={Diffusion-Based Probabilistic Modeling for Hourly Streamflow Prediction and Assimilation},
         author={Wencong Yang and Haoyu Ji and Leo Lonzarich and Yalan Song and Chaopeng Shen},
         year={2025},
         eprint={2510.08488},
         archivePrefix={arXiv},
         primaryClass={physics.geo-ph},
         url={https://arxiv.org/abs/2510.08488},
   }

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