Novel mathematical expression for dynamic stage-discharge relationship of Rivers under flow unsteadiness explored through machine learning models via symbolic regression in PySR

Stage-discharge (h-Q) rating curve (Q = ah^b) defined by parameters a and b is a crucial tool for measuring riverflow. However, this relationship struggles with flow fluctuations, backwater effects, tides, and changes in cross-sectional geometry, impacting its accuracy. Machine learning (ML) algorithms model the discharge with sequential stages as input, capturing intricate non-linear relationships, despite their “black box” nature that lacks explicit mathematical expressions. This study aims, therefore, to explore ML models trained on stage-discharge data using symbolic regression to produce actual mathematical expressions for stage-discharge relationships. By using sequential stage data as independent variables, the effect of unsteadiness is captured on stage-discharge relationships for River Churni (in India), Brays Bayou, Little Fishing Creek, and Obion (in USA). Symbolic regression using PySR tool is newly implemented to derive the mathematical expressions, followed by ML-based rating curve models. For each river, five generic mathematical expressions have been derived through symbolic regression where discharge is evaluated as polynomials of powered terms of sequential stages, rather than relying on conventional relationships. These expressions are assessed based on their modeling performance, prediction accuracy, and physical relevance. The results indicate that the coefficient of determination has significantly increased by 9–56 % with ML models compared to conventional relationships. The evaluation results show the best performance of new stage-discharge equation, recognizing the unique mathematical framework of stage-discharge relationship at river cross-sections and establishing a methodology for deriving it. This study executes a novel application of symbolic regression to evaluate explicit mathematical expressions relating to the sequential stage and discharge.