Chaves et al. (2026) When physics gets in the way: an entropy-based evaluation of conceptual constraints in hybrid hydrological models

Identification

• Journal: Hydrology and earth system sciences
• Year: 2026
• Date: 2026-02-04
• Authors: Manuel Álvarez Chaves, Eduardo Acuña Espinoza, Uwe Ehret, Anneli Guthke
• DOI: 10.5194/hess-30-629-2026

Research Groups

• Stuttgart Center for Simulation Science, Cluster of Excellence EXC 2075, University of Stuttgart, 70569 Stuttgart, Germany
• Institute of Water and Environment, Karlsruhe Institute of Technology (KIT), 76131 Karlsruhe, Germany

Short Summary

This study introduces an information theory-based metric to quantitatively evaluate the relative contributions of physics-based conceptual constraints and data-driven components in hybrid hydrological models. It finds that performance predominantly relies on the data-driven component, which often compensates for or even overwrites physics-based constraints, challenging the assumption that integrating physics inherently enhances model performance or interpretability.

Objective

• To introduce a quantitative metric based on Information Theory (entropy of LSTM hidden states) to evaluate the relative contributions of physics-based and data-driven components in hybrid hydrological models.
• To assess whether physics-based conceptual constraints genuinely enhance model performance and improve the representation of underlying physical processes, or if the data-driven component compensates for or overwrites these constraints.

Study Configuration

• Spatial Scale: Catchment scale; synthetic examples and a large-sample case study covering 671 river basins across England, Scotland, and Wales.
• Temporal Scale: Daily time series data; training (1 October 1980 to 31 December 1997), validation (1 October 1975 to 30 September 1980), and testing (1 January 1998 to 31 December 2008) periods. LSTM sequence length of 730 days.

Methodology and Data

• Models used:
  • Long Short-Term Memory (LSTM) networks (purely data-driven benchmark).
  • Hybrid models combining LSTM with conceptual hydrological models: Hybrid SHM (Simple Hydrological Model), Hybrid Bucket model, Hybrid Nonsense model.
  • Conceptual models with static parameters (SHM, Bucket, Nonsense) for comparison.
  • Variants: SHM with only one dynamic parameter, post-processing hybrid models.
  • Entropy-based metric: Kozachenko and Leonenko (KL) estimator for differential entropy of LSTM hidden states and conceptual model parameters.
• Data sources:
  • CAMELS-GB dataset: observed streamflow, 23 static catchment attributes (topography, soil, land cover, human influence, climate characteristics), and meteorological time-series data (catchment average precipitation, potential evapotranspiration, temperature).
  • Synthetic data generated from a "true" power reservoir model for didactic examples.

Main Results

• Hybrid models generally improve performance over purely conceptual models but do not exceed the performance of a pure LSTM in the CAMELS-GB case study.
• The data-driven (LSTM) component predominantly drives performance in hybrid models, often compensating for or effectively overwriting physics-based conceptual constraints.
• In didactic examples, the LSTM coupled with the "true" conceptual model showed minimal entropy in its hidden states and predicted static parameters, correctly recovering true values. Imperfect or over-parameterized constraints led to higher LSTM entropy and significant parameter variation, or parameters being driven to values that effectively simplify the model.
• In the CAMELS-GB case study, the pure LSTM exhibited the lowest entropy in 91% of basins, indicating that conceptual constraints generally increased the modeling challenge for the LSTM.
• Hybrid Nonsense surprisingly showed the lowest median entropy among hybrid models, suggesting its structure was easily transformed by the LSTM into a more suitable, parsimonious form.
• Visual analysis of time-varying parameters confirmed that the LSTM can effectively "morph" implausible conceptual structures (e.g., Nonsense model) into more efficient, single-storage-like systems by driving certain parameters to bypass storages.
• The study challenges the assumption that physics-informed machine learning necessarily preserves the physics as initially formulated, as the data-driven component may restructure imposed constraints.
• Entropy can be used to distinguish between equifinal models that achieve similar predictive performance but exhibit varying levels of LSTM activity and internal behavior modifications.

Contributions

• Introduction of a quantitative metric (based on information theory, specifically entropy of LSTM hidden states) to assess the relative contributions of data-driven and physics-based components in hybrid models.
• Demonstration of the metric's characteristics under synthetic conditions to guide understanding of physically meaningful versus problematic constraints.
• Suggestion of a diagnostic evaluation routine to understand the effective hybrid model's structure, beyond its prescribed conceptual model.
• Derivation of insights into the relative contribution of physics-based and data-driven components from a large-sample case study, illustrating how "physics may get in the way" under imperfectly known model settings.
• Uncovering the true role of constraints in presumably "physics-constrained" machine learning and guiding the development of more accurate representations of hydrological systems.

Funding

• Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2075 – 390740016
• Project 507884992
• University of Stuttgart (for open-access publication)

Citation

@article{Chaves2026When,
  author = {Chaves, Manuel Álvarez and Espinoza, Eduardo Acuña and Ehret, Uwe and Guthke, Anneli},
  title = {When physics gets in the way: an entropy-based evaluation of conceptual constraints in hybrid hydrological models},
  journal = {Hydrology and earth system sciences},
  year = {2026},
  doi = {10.5194/hess-30-629-2026},
  url = {https://doi.org/10.5194/hess-30-629-2026}
}