Physics-Informed Machine Learning for Short-Term Flood Prediction

Abstract: Accurate flood forecasting is essential for mitigating disaster risks and protecting communities. However, purely data-driven machine learning models often struggle in data-scarce environments and may violate fundamental hydrological principles. Standard Long Short-Term Memory (LSTM) networks can generate physically inconsistent predictions, particularly when extrapolating to extreme weather conditions. To address these limitations, we propose a Physics-Informed Machine Learning (PIML) framework that incorporates hydrological knowledge directly into the loss function of an LSTM model. Specifically, a Trend Alignment constraint penalizes directional inconsistencies between precipitation and discharge trends, improving model robustness without requiring complex hydrodynamic equations. This regularization encourages the model to learn physically plausible hydrograph behavior, even with limited training data, while enhancing reliability during peak flood events. Experimental results show that the proposed physics-informed model outperforms a standard LSTM baseline in data-scarce settings, increasing the Nash-Sutcliffe Efficiency (NSE) from 0.20 to 0.23 when trained on only 5% of the available data. Additional stress tests under simulated extreme climate scenarios demonstrate that the baseline model exhibits unstable behavior, whereas the physics-informed model maintains directional consistency and physical plausibility. Although accurately predicting extreme peak magnitudes remains challenging with limited data, the proposed approach substantially reduces unphysical fluctuations common in purely data-driven models. These findings demonstrate that simple physical constraints can significantly improve the reliability of deep learning models for real-time flood forecasting, offering a practical solution for ungauged basins and evolving climate conditions.

• The paper introduces a lightweight physics-informed LSTM that combines mean squared error with rainfall-runoff trend alignment and temporal smoothness penalties, improving physical consistency without solving explicit hydrological PDEs.
• The physics-informed model raises NSE from 0.2037 to 0.2321 with only 5% of training data, delivering an approximately 14% relative improvement, while full-data gains remain modest.
• The approach reduces erratic discharge responses under doubled precipitation but does not reliably resolve peak-flow magnitude, and its generalizability remains untested beyond one CAMELS-US basin.

Motivation and problem statement

Purely data-driven rainfall-runoff models, particularly LSTM networks, achieve strong statistical fit on gauged basins but lack physical grounding. When training data is scarce or when inputs fall outside the historical distribution—as during extreme precipitation events—such models can produce physically inconsistent discharge predictions, violating basic hydrological relationships such as the monotonic link between rainfall and rising runoff (2606.04143). The paper addresses this gap with a lightweight Physics-Informed Machine Learning (PIML) framework that embeds soft hydrological constraints directly into the LSTM loss function, avoiding the computational cost of solving explicit PDEs as in classical physics-informed neural network formulations (2606.04143).

Methodology

The framework is evaluated on a subset of the CAMELS-US dataset (Basin 01022500), using daily precipitation and discharge time series normalized via Min-Max scaling and segmented into 30-day lookback sequences. The baseline is a deliberately standard LSTM trained to minimize mean squared error (MSE), chosen so that any performance gain can be attributed to the physics-informed loss rather than architectural modifications.

The proposed model augments this objective with two penalty terms:

• Trend alignment: a hinge-style constraint enforcing that if precipitation occurs at time $t$ ($r_t > 0$), predicted discharge should not decrease over a lag window $K = 3$: $$\mathcal{L}_{\text{trend}} = \frac{1}{N}\sum_i \max(0,\; -\text{sign}(r_t)\cdot(\hat{y}_{t+K} - \hat{y}_t))$$.
• Temporal smoothness: penalizes squared differences between consecutive predicted discharges to suppress high-frequency fluctuations in the hydrograph.

The composite objective is $$\mathcal{L}_{\text{total}} = \mathcal{L}_{\text{data}} + 0.1\,\mathcal{L}_{\text{trend}} + 0.01\,\mathcal{L}_{\text{smooth}}$$, optimized with Adam. The loss weights and lag window are fixed hyperparameters rather than learned or basin-calibrated quantities.

Experimental design

Evaluation uses a chronological 80/20 train/test split with three metrics: RMSE for global accuracy, Nash-Sutcliffe Efficiency (NSE) for hydrological skill, and Peak Event MAPE computed on the top 5% of discharge events to assess reliability during flood peaks. Two stress conditions are introduced: a data-scarcity experiment restricted to 5% of training samples (78 samples), and an extreme-climate test in which precipitation inputs are artificially doubled.

Results

Under the full-data regime, gains from the physics-informed loss are marginal: NSE improves from 0.2441 to 0.2558, RMSE decreases from 4.1410 to 4.1090 mm/d, and Peak MAPE improves slightly from 39.97% to 39.73%. The authors interpret this as evidence that soft physical constraints do not degrade—and can slightly enhance—high-flow capture when data is abundant.

Regime	Model	RMSE (mm/d)	NSE	Peak MAPE
Full data	Baseline LSTM	4.14	0.24	39.97%
Full data	Physics-informed	4.11	0.26	39.73%
5% data	Baseline LSTM	—	0.2037	—
5% data	Physics-informed	—	0.2321	—

The more consequential result appears under data scarcity: with only 78 training samples, the baseline degrades to an NSE of 0.2037 while the physics-informed model retains 0.2321, a relative improvement of roughly 14%. This supports the central claim that embedded physical laws act as regularizers precisely where statistical supervision is insufficient. In the doubled-precipitation stress test, the physics-constrained model produces directionally consistent discharge rises, whereas the baseline exhibits saturation and erratic fluctuations—an out-of-distribution failure mode that trend alignment mitigates.

Two caveats temper these findings. First, absolute performance levels are modest: an NSE of approximately 0.26 on full data is well below state-of-the-art rainfall-runoff benchmarks, suggesting the single-basin setup or short lookback configuration limits overall skill. Second, the authors concede that capturing the exact magnitude of extreme peaks remains unresolved in sparse regimes; the method eliminates unphysical fluctuations but does not solve peak-magnitude accuracy, which is arguably the most operationally critical quantity for flood warning.

Limitations and open questions

The study is confined to a single CAMELS-US basin, so cross-basin generalization of the fixed loss weights ($0.1$, $0.01$) and lag window ($K=3$) remains untested. The constraints are heuristic rather than derived from conservation laws, leaving open whether they encode sufficient hydrological structure for basins with different runoff regimes—for example snowmelt-dominated catchments where the rain-rise assumption may not hold. The authors also leave open adaptive estimation of basin-specific parameters such as $K$, integration into modern recurrent architectures beyond the standard LSTM, and hybridization with process-based models. Finally, because the extreme-climate test relies on synthetic input doubling rather than observed events, real-world extrapolation behavior under non-stationarity is not directly validated.

Conclusion

This paper demonstrates that simple, targeted soft constraints—trend alignment between precipitation and discharge, plus temporal smoothness—can be appended to a standard LSTM at negligible cost and yield measurable robustness gains in data-scarce and out-of-distribution settings, including a 14% relative NSE improvement at 5% training data and elimination of erratic behavior under synthetic extreme precipitation. The contribution is less about peak predictive accuracy, which remains modest, than about establishing that lightweight physical regularization provides a practical safeguard against physically implausible predictions in ungauged or changing basins.