Articles | Volume 28, issue 13
https://doi.org/10.5194/hess-28-3051-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
https://doi.org/10.5194/hess-28-3051-2024
© Author(s) 2024. This work is distributed under
the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Research article
| 
15 Jul 2024
Research article |
| 15 Jul 2024
| 15 Jul 2024
When ancient numerical demons meet physics-informed machine learning: adjoint-based gradients for implicit differentiable modeling
Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA
Wouter J. M. Knoben
Department of Civil Engineering, Schulich School of Engineering, University of Calgary, Calgary, Alberta, Canada
Martyn P. Clark
Department of Civil Engineering, Schulich School of Engineering, University of Calgary, Calgary, Alberta, Canada
Dapeng Feng
Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA
Department of Earth System Science, Stanford University, Stanford, CA, USA
Kathryn Lawson
Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA
Kamlesh Sawadekar
Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA
Civil and Environmental Engineering, The Pennsylvania State University, University Park, PA, USA
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- Update hydrological states or meteorological forcings? Comparing data assimilation methods for differentiable hydrologic models A. Jamaat et al. https://doi.org/10.1016/j.jhydrol.2025.134137
- Improving differentiable hydrologic modeling with interpretable forcing fusion K. Sawadekar et al. https://doi.org/10.1016/j.jhydrol.2025.133320
- Two-dimensional differential form of distributed Xinanjiang model J. Zhao et al. https://doi.org/10.5194/hess-29-3745-2025
- Deep learning for the probabilistic prediction of semi-continuous hydrological variables – An application to streamflow prediction across CONUS J. Quilty & M. Jahangir https://doi.org/10.1016/j.jhydrol.2026.134986
- Agentic AI for Reservoir Flood Dispatching: A Physics–Cognition Collaborative Framework S. Yan & S. Hao https://doi.org/10.3390/app16168171
- A flexible, differentiable framework for neural-enhanced hydrological modeling: Design, implementation, and applications with HydroModels.jl X. Jing et al. https://doi.org/10.1016/j.envsoft.2025.106802
- Evaluating the Hydrological Applicability of Satellite Precipitation Products Using a Differentiable, Physics-Based Hydrological Model in the Xiangjiang River Basin, China S. Yan et al. https://doi.org/10.3390/rs18010137
- δHT4P: A differentiable physical modeling framework for thermal evolution of permafrost L. Gou & W. Likos https://doi.org/10.1016/j.compgeo.2026.108132
- Coupling SWAT+ with LSTM for enhanced and interpretable streamflow estimation in arid and semi-arid watersheds, a case study of the Tagus Headwaters River Basin, Spain S. Asadi et al. https://doi.org/10.1016/j.envsoft.2025.106360
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- A Robust Calibration and Evaluation Framework for Dynamic Catchment Characteristics in Hydrological Modeling T. Lan et al. https://doi.org/10.5194/hess-30-2455-2026
- AI Solutions for Improving Sustainability in Water Resource Management J. Silva https://doi.org/10.3390/su18042154
- DMFS: differentiable modeling for frozen soil thermodynamic characteristics Y. Ren et al. https://doi.org/10.1139/cgj-2025-0364
- Protocols for Water and Environmental Modeling Using Machine Learning in California M. He et al. https://doi.org/10.3390/hydrology12030059
- Rapid 2D hydrodynamic flood modeling using deep learning surrogates F. Haces-Garcia et al. https://doi.org/10.1016/j.jhydrol.2024.132561
- When physics gets in the way: an entropy-based evaluation of conceptual constraints in hybrid hydrological models M. Álvarez Chaves et al. https://doi.org/10.5194/hess-30-629-2026
- A hybrid physics–AI approach using universal differential equations with state-dependent neural networks for learnable, regionalizable, spatially distributed hydrological modeling N. Huynh et al. https://doi.org/10.5194/gmd-19-1055-2026
Saved (final revised paper)
Latest update: 03 Sep 2026
Short summary
Differentiable models (DMs) integrate neural networks and physical equations for accuracy, interpretability, and knowledge discovery. We developed an adjoint-based DM for ordinary differential equations (ODEs) for hydrological modeling, reducing distorted fluxes and physical parameters from errors in models that use explicit and operation-splitting schemes. With a better numerical scheme and improved structure, the adjoint-based DM matches or surpasses long short-term memory (LSTM) performance.
Differentiable models (DMs) integrate neural networks and physical equations for accuracy,...
