Metaheuristic Optimized ANFIS Model for Data Driven Prediction of Scour Depth Downstream of Ski Jump Spillways

Accurate prediction of scour depth downstream of ski‑jump spillways is essential for dam safety, optimal plunge pool design, and cost‑effective protection, due to complex nonlinear hydraulic interactions governing erosion processes. This study introduces a hybrid Adaptive Neuro‑Fuzzy Inference System optimized by the Harris Hawks Optimization algorithm (ANFIS–HHO) for predicting the relative scour depth \frac{d_s}{d_w} downstream of ski‑jump spillways. The model performance is quantitatively compared with regression‑based equations derived from dimensional analysis. Tailwater depth (dw) discharge per unit width (q), gravitational acceleration (g), total head (H1), bucket radius (R), median sediment size (d50), and lip angle () were used to extract dimensionless parameters, namely \frac{\mathrm{q}}{\sqrt{\mathrm{g}\mathrm{d}_\mathrm{w}^\mathrm{3}}}, \frac{\mathrm{H}_\mathrm{1}}{\mathrm{d}_\mathrm{w}}, \frac{\mathrm{R}}{\mathrm{d}_\mathrm{w}}, \frac{\mathrm{d}_{\mathrm{50}}}{\mathrm{d}_\mathrm{w}} and \mathrm{\varphi}. The ANFIS–HHO model achieved excellent performance with RMSE=0.01794, MAE= 0.01060, \mathrm{(}\frac{\mathrm{d}_\mathrm{s}}{\mathrm{d}_\mathrm{w}}\mathrm{)}_{{\mathrm{DDR}}_{\mathrm{max}}}\mathrm{=2.945} and R2=0.99234 in training, and RMSE=0.02698, MAE=0.02161, \mathrm{(}\frac{\mathrm{d}_\mathrm{s}}{\mathrm{d}_\mathrm{w}}\mathrm{)}_{{\mathrm{DDR}}_{\mathrm{max}}}\mathrm{=1.437} and R2= 0.98725 in testing. In contrast, the regression model provided lower but acceptable accuracy (calibration: RMSE=0.73094, MAE=0.58638, \mathrm{(}\frac{\mathrm{d}_\mathrm{s}}{\mathrm{d}_\mathrm{w}}\mathrm{)}_{{\mathrm{DDR}}_{\mathrm{max}}}\mathrm{=1.019}, R2= 0.89764; verification: RMSE= 0.52440, MAE=0.43286, \mathrm{(}\frac{\mathrm{d}_\mathrm{s}}{\mathrm{d}_\mathrm{w}}\mathrm{)}_{{\mathrm{DDR}}_{\mathrm{max}}}\mathrm{=4.56}, R2=0.98236). Although the hybrid model demonstrates superior predictive capability, the regression equation offers the advantage of being a white‑box model, enabling direct and interpretable estimation of scour depth for practical hydraulic design applications.