The Differentiable Distributed Regression Model (dDRM) Balancing Explainability and Predictive Performance

The field of hydrology has undergone significant transformation over the past decade, driven by advancements in machine learning and data-driven techniques. A key breakthrough came from the work of Kratzert et al. (2018), who demonstrated that purely data-driven LSTM models could outperform traditional hydrological models in over 600 catchments across North America. However, while these models significantly improve predictive performance, they often sacrifice interpretability and explainability.

To address this trade-off, researchers have explored new approaches that merge physical principles with data-driven methods. One promising innovation is the concept of differentiable modeling, introduced by Chen et al. in 2022. This approach transforms physical models into differentiable functions, allowing neural networks to represent and learn model parameters. By doing so, differentiable modeling enhances flexibility while maintaining a foundation in physical principles.

This research presents a novel differentiable hydrological model called the Differentiable Distributed Regression Model (dDRM). The dDRM builds on the principles of differentiable modeling with the structure of a conceptually lumped model using a simplified representation of physics ("smooth" HBV model). Inspired by the simplicity of the LSTM model, which aggregates data at the catchment level rather than relying on a grid-based representation, we introduce four equally sized elevation zones instead of grid cells in the dDRM. These zones inherently reflect differences in hydrological processes, such as precipitation, temperature, and snowmelt dynamics, enabling the model to account for spatial heterogeneity while maintaining computational efficiency.

By leveraging the principles of differentiable modeling, the dDRM achieves a balance between explainability and predictive performance. To evaluate model performance, we tested the dDRM across sixty-three catchments in southern Norway, in a gauged setting. Only precipitation and temperature were used as input data. For benchmarking purposes, we also trained an LSTM model to the same catchments. 

Our results demonstrate that the dDRM outperforms the fine-tuned LSTM model in both daily predictions and cumulative runoff volumes. These findings underscore the potential of differentiable hydrological models to bridge the gap between performance and interpretability. By combining physical principles with data-driven techniques, the dDRM provides a pathway toward more effective and understandable forecasting tools in hydrology.