# Differentiable, learnable, regionalized process-based models with physical outputs can approach state-of-the-art hydrologic prediction accuracy

arXiv: https://arxiv.org/abs/2203.14827 · HTML: https://mlsanity.com/paper/2203.14827/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 87 references that are in the map.
- First posted on arXiv 28 Mar 2022.

## Builds on (most cited first, 9 of 9)

- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Towards Learning Universal, Regional, and Local Hydrological Behaviors via Machine-Learning Applied to Large-Sample Datasets](https://mlsanity.com/paper/1907.08456/) (2019)
- [A trans-disciplinary review of deep learning research for water resources scientists](https://mlsanity.com/paper/1712.02162/) (2017)
- [Enhancing streamflow forecast and extracting insights using long-short term memory networks with data integration at continental scales](https://mlsanity.com/paper/1912.08949/) (2019)
- [From calibration to parameter learning: Harnessing the scaling effects of big data in geoscientific modeling](https://mlsanity.com/paper/2007.15751/) (2020)
- [The data synergy effects of time-series deep learning models in hydrology](https://mlsanity.com/paper/2101.01876/) (2021)
- [Prediction in ungauged regions with sparse flow duration curves and input-selection ensemble modeling](https://mlsanity.com/paper/2011.13380/) (2020)
- [Prolongation of SMAP to Spatio-temporally Seamless Coverage of Continental US Using a Deep Learning Neural Network](https://mlsanity.com/paper/1707.06611/) (2017)

## Built on by (most cited first, 21 of 21)

- [Knowledge-guided Machine Learning: Current Trends and Future Prospects](https://mlsanity.com/paper/2403.15989/) (2024)
- [Sensitivity-Constrained Fourier Neural Operators for Forward and Inverse Problems in Parametric Differential Equations](https://mlsanity.com/paper/2505.08740/) (2025)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Probing the limit of hydrologic predictability with the Transformer network](https://mlsanity.com/paper/2306.12384/) (2023)
- [Time Series Predictions in Unmonitored Sites: A Survey of Machine Learning Techniques in Water Resources](https://mlsanity.com/paper/2308.09766/) (2023)
- [Towards Interpretable Physical-Conceptual Catchment-Scale Hydrological Modeling using the Mass-Conserving-Perceptron](https://mlsanity.com/paper/2401.14521/) (2024)
- [Differentiable modeling to unify machine learning and physical models and advance Geosciences](https://mlsanity.com/paper/2301.04027/) (2023)
- [Using Machine Learning to Discover Parsimonious and Physically-Interpretable Representations of Catchment-Scale Rainfall-Runoff Dynamics](https://mlsanity.com/paper/2412.04845/) (2024)
- [A Deep State Space Model for Rainfall-Runoff Simulations](https://mlsanity.com/paper/2501.14980/) (2025)
- [HydroAgent: Closing the Gap Between Frontier LLMs and Human Experts in Hydrologic Model Calibration via Simulator-Grounded RL](https://mlsanity.com/paper/2605.17792/) (2026)
- [Learning Regionalization using Accurate Spatial Cost Gradients within a Differentiable High-Resolution Hydrological Model: Application to the French Mediterranean Region](https://mlsanity.com/paper/2308.02040/) (2023)
- [Update hydrological states or meteorological forcings? Comparing data assimilation methods for differentiable hydrologic models](https://mlsanity.com/paper/2502.16444/) (2025)
- [Distinct hydrologic response patterns and trends worldwide revealed by physics-embedded learning](https://mlsanity.com/paper/2504.10707/) (2025)
- [Scientifically-Interpretable Reasoning Network (ScIReN): Discovering Hidden Relationships in the Carbon Cycle and Beyond](https://mlsanity.com/paper/2506.14054/) (2025)
- [Physics-guided Emulators Reveal Resilience and Fragility under Operational Latencies and Outages](https://mlsanity.com/paper/2510.18535/) (2025)
- [Leveraging Exogenous Signals for Hydrology Time Series Forecasting](https://mlsanity.com/paper/2511.11849/) (2025)
- [Topology enables learning-based hydrodynamic prediction of the global river system](https://mlsanity.com/paper/2602.22293/) (2026)
- [Process-Aware AI for Rainfall-Runoff Modeling: A Mass-Conserving Neural Framework with Hydrological Process Constraints](https://mlsanity.com/paper/2603.25093/) (2026)
- [A Differentiable Framework for Global Circulation Model Precipitation Bias Correction](https://mlsanity.com/paper/2604.23045/) (2026)
- [HydroAgent: Formalizing Forecaster Expertise into Skill-Orchestrated Flood Forecasting Workflows](https://mlsanity.com/paper/2607.23983/) (2026)
- [Hapi: A Multivariable Land-Surface Transformer for Medium-Range Hydrological Forecasting at Continental Scale](https://mlsanity.com/paper/2609.22702/) (2026)

## Abstract (arXiv)

Predictions of hydrologic variables across the entire water cycle have significant value for water resource management as well as downstream applications such as ecosystem and water quality modeling. Recently, purely data-driven deep learning models like long short-term memory (LSTM) showed seemingly-insurmountable performance in modeling rainfall-runoff and other geoscientific variables, yet they cannot predict untrained physical variables and remain challenging to interpret. Here we show that differentiable, learnable, process-based models (called δ models here) can approach the performance level of LSTM for the intensively-observed variable (streamflow) with regionalized parameterization. We use a simple hydrologic model HBV as the backbone and use embedded neural networks, which can only be trained in a differentiable programming framework, to parameterize, enhance, or replace the process-based model modules. Without using an ensemble or post-processor, δ models can obtain a median Nash Sutcliffe efficiency of 0.732 for 671 basins across the USA for the Daymet forcing dataset, compared to 0.748 from a state-of-the-art LSTM model with the same setup. For another forcing dataset, the difference is even smaller: 0.715 vs. 0.722. Meanwhile, the resulting learnable process-based models can output a full set of untrained variables, e.g., soil and groundwater storage, snowpack, evapotranspiration, and baseflow, and later be constrained by their observations. Both simulated evapotranspiration and fraction of discharge from baseflow agreed decently with alternative estimates. The general framework can work with models with various process complexity and opens up the path for learning physics from big data.