A Machine Learning-Based Approach for Precise Estimation of Discharge Coefficient in Labyrinth and Piano Key Weirs

Nonlinear weirs are widely used to increase spillway discharge capacity under constrained footprints; however, accurate estimation of the discharge coefficient (Cd) remains challenging due to highly nonlinear hydraulic behavior. This study experimentally and numerically evaluates the hydraulic performance of rectangular piano key weirs (RPKW) and rectangular labyrinth weirs (RLW) using physical modeling and advanced machine‑learning techniques. A total of 90 steady‑flow laboratory experiments were conducted, covering relative crest length (L/B=0.8–1.2), relative width (W/B=0.2–0.4), and relative upstream head (HT/P<=0.6 ). Experimental results showed that Cd initially increases with and then decreases due to flow interference, while increasing and significantly enhances discharge efficiency. A 1.5‑fold increase in resulted in approximately 43% and 25% increases in Cd for RPKW and RLW, respectively. For predictive modeling, regression, Gene Expression Programming (GEP), and a hybrid Particle Swarm Optimization–GEP (PSO‑GEP) approach were developed and evaluated using RMSE, MAE, coefficient of determination (R2), and the Developed Discrepancy Ratio (DDR). The PSO‑GEP model exhibited superior performance, achieving (R2=0.988) and RMSE ≤ 0.0023 in the testing phase for both weir types. Moreover, the highest Cd(DDR)max values (12.58 for RPKW and 13.03 for RLW) were obtained by PSO‑GEP, indicating enhanced reliability in predicting extreme discharge conditions. The proposed hybrid framework provides accurate, robust, and explicit predictive formulations suitable for practical spillway design and optimization.