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Enhancing predictive skills in physically-consistent way: Physics Informed Machine Learning for Hydrological Processes

Pravin Bhasme, Jenil Vagadiya, Udit Bhatia

arXiv:2104.11009v1stat.MLcs.LGstat.AP

TL;DR

Hydrological modeling must balance predictive skill with physical consistency and process understanding. This paper embeds machine-learning algorithms in the abcd model structure to predict monthly streamflow and actual evapotranspiration. In the Narmada basin, the authors report stronger predictive performance than pure physics-based or ML approaches, reduced uncertainty bounds, and consistent annual water-budget performance.

  • Problem

    Physics-based models provide process understanding but can have rigid structures, while ML models can predict well but may ignore physical constraints.

    Source: §1 · p. 2

  • Method

    The PIML framework replaces relationships in the abcd model with ML functions while retaining its covariate structure to estimate actual evapotranspiration and streamflow.

    Source: §1 · p. 4

  • Results

    For the study area, PIML showed high prediction performance compared with pure physics-based or ML approaches, reduced uncertainty bounds for both variables, and consistent annual water-budget performance across variants.

    Source: §5 · p. 15

  • Takeaways & Limitations

    The framework provides a way to add flexibility to rigid hydrological model structures and could extend to conceptual models with more intermediate steps.

    Source: §5 · p. 15

    The evaluation is limited to a single hydrological unit and monthly time series; distributed and semi-distributed structures and daily or sub-daily steps remain to be validated.

    Source: §5 · p. 15

Abstract

from arXiv · show

Current modeling approaches for hydrological modeling often rely on either physics-based or data-science methods, including Machine Learning (ML) algorithms. While physics-based models tend to rigid structure resulting in unrealistic parameter values in certain instances, ML algorithms establish the input-output relationship while ignoring the constraints imposed by well-known physical processes. While there is a notion that the physics model enables better process understanding and ML algorithms exhibit better predictive skills, scientific knowledge that does not add to predictive ability may be deceptive. Hence, there is a need for a hybrid modeling approach to couple ML algorithms and physics-based models in a synergistic manner. Here we develop a Physics Informed Machine Learning (PIML) model that combines the process understanding of conceptual hydrological model with predictive abilities of state-of-the-art ML models. We apply the proposed model to predict the monthly time series of the target (streamflow) and intermediate variables (actual evapotranspiration) in the Narmada river basin in India. Our results show the capability of the PIML model to outperform a purely conceptual model ($abcd$ model) and ML algorithms while ensuring the physical consistency in outputs validated through water balance analysis. The systematic approach for combining conceptual model structure with ML algorithms could be used to improve the predictive accuracy of crucial hydrological processes important for flood risk assessment.

1 Introduction

Hydrological modeling faces a tension between machine-learning prediction and physics-based interpretability, while either approach alone can leave important challenges unresolved. The paper proposes embedding machine-learning algorithms within the abcd model structure to predict streamflow and intermediate variables.

1 Introduction

  • Machine-learning models can predict catchment responses but face limited interpretability, physical inconsistencies, and equifinality, while physics-based models depend critically on parameter calibration.The introduction questions whether process understanding that does not improve prediction, or prediction without physical realism, is sufficient.Source: §1 · p. 2
  • Prior hybrid approaches span multiple scientific domains, including physics-guided architectures and constraints for fluid, seismic, and lake-temperature modeling.The introduction notes that earth-science applications also need to preserve spatiotemporal structure and represent prediction uncertainty.Source: §1 · pp. 2–3
  • Earlier hydrological hybrids modeled some intermediate variables, but the cited SWAT-emulating architecture omitted many fluxes and state variables, limiting output interpretability.The omitted variables include surface runoff, groundwater flow, percolation, and aquifer storage.Source: §1 · p. 3
  • The proposed PIML framework replaces abcd empirical relationships at multiple stages with ML algorithms, using the model structure to select inputs, intermediate variables, and streamflow outputs.It is applied to the Narmada River Basin at Sandia and compared with conceptual and pure ML approaches.Source: §1 · pp. 3–4

2 Methods

The methods combine the abcd model’s hydrological structure with machine-learning algorithms and evaluate predictions using standard hydrological metrics. The framework estimates actual evapotranspiration and streamflow while using water-balance-informed covariates.

2.1 Review of the abcd model

  • The abcd model is a parsimonious catchment-scale water-balance model with soil-moisture and groundwater storage compartments, driven by precipitation and potential evapotranspiration.Its parameterized relationships connect hydrological variables, while mass-balance equations describe the storage compartments.Source: §2.1 · pp. 4–5

2.2 Review of Machine Learning Algorithms

  • The reviewed ML methods include LSTMs for temporal dependencies, Bayesian networks for uncertainty bounds, GPR and SVR for regression, and LASSO and Ridge for regularization.LSTMs use gates and memory cells; SVR uses a kernel trick to learn nonlinear relationships.Source: §2.2 · pp. 5–6

2.3 Physics Informed Machine Learning model

  • PIML uses the abcd model’s covariate structure to replace rigid relationships with ML functions, first estimating actual evapotranspiration and then streamflow.The ML functions determine the relationships, while the conceptual structure guides covariate choice and preserves an interpretable sequence.Source: §2.3 · pp. 6–7, Figure 1 · p. 7

2.4 Evaluation metrics

  • Model performance is assessed with NSE, PBIAS, and RMSE to measure efficiency, prediction bias, and output error, respectively.The methods use established hydrological evaluation metrics.Source: §2.4 · p. 7

2.4.1 Nash-Sutcliffe Efficiency

  • NSE assesses goodness of fit on a scale from negative infinity to 1, with 1 indicating a perfect match and values below 0 favoring the observed mean.The observed mean is a better predictor than the model output when NSE is below zero.Source: §2.4.1 · p. 8

2.4.2 Percent Bias

  • PBIAS measures how well the model estimates average output magnitudes; zero is optimal, while positive and negative values indicate underprediction and overprediction.Source: §2.4.2 · p. 8

2.4.3 Root Mean Square Error

  • RMSE measures deviations between modeled and observed data, with zero indicating a perfect match.Source: §2.4.3 · p. 8
  • For monthly simulations, NSE above 0.8 is very good, while values at or below 0.5 are unsatisfactory; negative NSE is unacceptable.The criteria also classify 0.7 < NSE ≤ 0.8 as good and 0.5 < NSE ≤ 0.7 as satisfactory.Source: §2.4.3 · p. 8

3 Study area and datasets

The study focuses on the Narmada basin upstream of Sandia and combines meteorological, streamflow, and land-surface data for monthly modeling. Potential evapotranspiration is derived from daily temperature and radiation data, and models are evaluated on a held-out 2009–2014 period.

3 Study area and datasets

  • The study area upstream of Sandia covers 38,571 km2 within the 1,312 km Narmada River basin, which drains 98,796 km2.Sandia gauge station is at 22.92°N, 78.35°E.Source: §3 · p. 8
  • Daily precipitation and minimum and maximum temperatures come from IMD for 1979–2014 at 0.25° and 1° spatial resolutions, respectively.Observed Sandia streamflow comes from India-WRIS; daily soil moisture, groundwater storage, and evapotranspiration come from GLDAS Catchment Land Surface Model L4.Source: §3 · p. 8
  • Daily potential evapotranspiration is calculated with the Hargreaves method, summed to monthly values, then spatially averaged across the study region.PET is expressed in mm/day; the passage defines Ra as extraterrestrial radiation and Tmax, Tmin, and Tavg as surface temperatures in °C.Source: §3 · pp. 8–9
  • Models use 1979–2008 for training and 2009–2014 for testing, with 1976–1978 as the abcd model’s warm-up period.The same training and testing periods are used for the abcd, ML, and PIML models, and metrics are calculated on the test set.Source: §3 · p. 9

4 Results

In the Narmada-basin tests, the conceptual abcd model could fit streamflow while failing on intermediate variables, and standalone ML performance varied with the input set. PIML improved predictions of streamflow and actual ET over the compared approaches, and its annual water-balance results were consistent with physical constraints.

4.1 Evaluating performance of the physics-based model

  • The abcd model’s all-variable calibration was unacceptable for streamflow and other predicted variables; calibrating against streamflow and ET still left ET unacceptable.The model’s groundwater time series also showed a significant offset from observations across cases.Source: §4.1 · pp. 9–10, Figure 2 · p. 10, Table 2 · p. 11

4.2 Performance evaluation of ML Algorithms

  • With precipitation and temperature inputs, LSTM, LASSO, Ridge, and Bayesian LSTM were satisfactory, while SVR and GPR were unsatisfactory; all tested models were satisfactory with precipitation and ET inputs by NSE.Bayesian LSTM had higher PBIAS in the latter input case, alongside improved NSE and decreased RMSE; changing its predictor set did not reduce uncertainty in Figure 4.Source: §4.2 · p. 11, Table 3 · p. 11, Figure 4 · p. 12

4.3 Performance Evaluation of PIML model

  • PIML variants improved monthly streamflow predictions over the corresponding ML models, with abcd+GPR and abcd+LSTM also outperforming the abcd model; five variants outperformed abcd for ET.For streamflow, PIML+LSTM and PIML+GPR were rated “very good”; for ET, PIML+SVR and PIML+GPR were rated “very good.”Source: §4.3 · pp. 12–13, Table 4 · p. 13
  • The Bayesian-LSTM PIML variant produced 90% confidence intervals for both ET and streamflow, with narrower streamflow uncertainty bounds than pure ML algorithms.The PIML time-series plots show most variants capturing trends and magnitudes reasonably for both variables.Source: §4.3 · pp. 13–14, Figure 6 · p. 14, Figure 5 · p. 14
  • The hybrid SVR-for-ET/GPR-for-Q variant had a -1.893% annual water-balance deviation, compared with -3.255% for SVR and -6.754% for GPR.The study reports an initial -9.884 mm difference between precipitation and ET plus streamflow; soil-moisture and groundwater storage changes were treated as negligible in the balance check.Source: §4.3 · p. 13

5 Conclusion

The proposed PIML model combines conceptual-model process structure with ML to predict hydrological variables. For the study, it showed high prediction performance, reduced uncertainty bounds, and consistent annual water-budget performance across variants.

5 Conclusion

  • The study demonstrates monthly prediction of actual evapotranspiration and streamflow for a single hydrological unit.Figure 7 presents observed and modeled time series for evapotranspiration and streamflow.Source: §5 · pp. 14–15, Figure 7 · p. 15
  • PIML outputs showed high prediction performance versus pure physics-based and ML approaches, reduced uncertainty bounds for both variables, and consistent annual water-budget performance across variants.The annual water-balance analysis is reported for the testing period.Source: §5 · p. 15, Table 5 · p. 15
  • The model captures complex, nonlinear dependencies while following the logical sequence guided by the conceptual hydrological model.The approach replaces empirical equations in the abcd model with ML relationships between input, intermediate, and target variables.Source: §5 · pp. 14–15
  • The framework, demonstrated with the abcd water-balance model, could extend to other conceptual models with more intermediate steps.Source: §5 · p. 15
  • Validation is still needed for distributed and semi-distributed structures and daily or sub-daily time steps; upstream reservoirs were omitted and may affect skill, especially at those finer scales.Source: §5 · p. 15
  • Performance is highly sensitive to the ML algorithms selected for intermediate steps, and no proven selection guidelines are available.Source: §5 · p. 16
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