Machine learning model for stage-discharge curve calculation

Stage-discharge relationships (rating curves) are fundamental in hydrology but remain challenging to establish in experimental catchments, where observations are sparse, irregular, and uncertain. Conventional regression models provide simple and interpretable solutions, yet often fail to capture nonlinearities in hydraulically complex environments. Purely data-driven machine learning (ML) models offer flexibility, but their performance deteriorates under data scarcity and they often produce physically implausible results. We present a hybrid physics-informed machine learning (PIML) framework that integrates a log-log regression baseline with residual corrections from Support Vector Regression (SVR) and Multilayer Perceptron (MLP) models. By embedding hydrological constraints such as monotonicity, non-negativity, and continuity, the framework ensures physically consistent rating curves while leveraging ML to capture nonlinear deviations. The approach was developed in four contrasting catchments and validated across 20 independent evaluation sites. Results show that both hybrid models outperform conventional regression, with the Hybrid MLP consistently providing the most accurate and generalizable predictions (median R² and NSE > 0.98) even when calibrated with as few as 8–15 discharge measurements. The framework is particularly effective in irregular or hydraulically complex basins, while differences to conventional regression are minimal in stable profiles. These findings demonstrate that PIML enables systematic, transferable, and reproducible rating curve development under sparse and uncertain data conditions. The framework offers a practical alternative to subjective or ad hoc methods, advancing discharge estimation in experimental hydrology and supporting applications in data-limited and hydraulically complex environments.

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