\(\delta\) HBV1.1p-hydroDL¶

Code Release¶

The code for differentiable HBV model 1.1p is released at this zenodo link. \(\delta\) HBV1.1 has also been integrated into 𝛿MG, our new highly user-friendly platform built as a generic differentiable modeling framework, where the workflow is much more streamlined and unified. It is recommended new users get started on 𝛿MG. δMG: https://github.com/mhpi/generic_deltamodel/tree/master δHBV1.1p in δMG: https://github.com/mhpi/generic_deltamodel/blob/master/example/hydrology/example_dhbv_1_1p.ipynb

Summary¶

Recently, a hybrid framework combining machine learning (ML) and process-based equations, termed differentiable modeling, has shown comparable accuracy to pure ML models while offering enhanced interpretability and spatial generalizability. However, it remained unclear how well hybrid models generalize to extreme floods outside of the range of training data, and whether optimizing models for extreme events jeopardizes spatial generalizability and the physical significance of internal variables. Here we evaluated multiple versions of a differentiable model (δHBV1.0 and δHBV1.1p) for predicting unseen extreme events, and benchmarked them against a widely-applied long short-term memory (LSTM) network on the CAMELS data set. We found that both δHBV and LSTM models performed well, with δHBV1.1p outperforming LSTM for events with a return period of 5 years or more. This advantage was more pronounced as the return period increased (0.06 higher median Nash-Sutcliffe efficiency and lower peak flow errors for 80% of the 50-year or rarer events). Loss function choice had a larger impact on δHBV1.1p than on LSTM, and we showed the proper loss led to δHBV models that further surpassed LSTM in different performance aspects. Furthermore, allowing more dynamic parameters improved the extreme metrics, had no negative impact on spatial generalization, and exerted a minimal influence on the untrained variables. We hypothesize that δHBV's mass balance and first-order exchange terms help constrain and inform its responses to mitigate the underestimation of peaks compared to LSTM. We conclude that adopting interpretable structural priors can improve generalizability to unseen cases and thus increase model reliability to better inform stakeholder preparedness.

Bibtex Citation¶

@article{song2026physics,
  title={Physics-informed, Differentiable hydrologic models for capturing unseen extreme events},
  author={Song, Yalan and Sawadekar, Kamlesh and Frame, Jonathan M and Pan, Ming and Clark, Martyn P and Knoben, Wouter JM and Wood, Andrew W and Lawson, Kathryn E and Patel, Trupesh and Shen, Chaopeng},
  journal={Water Resources Research},
  volume={62},
  number={2},
  pages={e2025WR040414},
  year={2026},
  publisher={Wiley Online Library}
}