Hybrid machine learning models with particle swarm optimization based tuning for solitary wave runup prediction on sloping beaches

Abstract

Accurate prediction of maximum wave runup, R, is central to coastal flooding assessment, freeboard specification, and risk-informed design. However, compact solitary-wave runup relations may exhibit slope-dependent bias when applied beyond their calibration envelope. This study proposes a scaling-informed hybrid residual-learning framework to predict normalized runup (R/d) where d is the still-water depth, using two reduced predictors: relative wave amplitude ε = A/d and beach slope s. A parsimonious power-law regression derived from reduced-variable scaling provides the power-law baseline, while four residual correctors Gaussian Process Regression (GPR), Support Vector Regression (SVR), Extreme Learning Machine (ELM), and Symbolic Regression (SR) learn a multiplicative correction in log-space. Hyperparameters are optimized via Particle Swarm Optimization (PSO) within each training fold, and cross-slope transferability is assessed using a strict leave-one-slope-out (LOSO) protocol. Relative to the LOSO power-law baseline (R²=0.932, RMSE = 0.0838, MAPE = 7.21%), the best hybrid model (SR-PSO) improves performance to R 2 = 0.951, RMSE = 0.0724 and MAPE = 5.82%, corresponding to about 13.6% lower RMSE and 18.2% lower MAPE. Beyond point accuracy, SR-PSO yields an explicit closed-form correction suitable for engineering use, while GPR-PSO provides predictive intervals for uncertainty-aware screening. Overall, the proposed framework improves cross-slope transferability while retaining a compact and interpretable structure.