Improved predictive formulae for wave overtopping at sloped breakwaters using interpretable machine learning models

Accurate prediction of mean wave overtopping discharge is essential for the safe and cost-effective design of coastal defence structures. While traditional empirical, physical, and numerical models remain important, Machine Learning (ML) has recently emerged as a powerful complementary tool. This study presents a ML–based framework to predict mean wave overtopping discharge at sloped breakwaters, with a focus on both predictive accuracy and model interpretability, supported by a series of structured pre- and post-processing steps. Five ML algorithms were evaluated: two decision tree–based models, i.e., Random Forest (RF) and Gradient Boosted Decision Trees (GBDT), and three kernel-based models, i.e., Artificial Neural Networks (ANN), Support Vector Regression (SVR), and Gaussian Process Regression (GPR). The models were trained and validated using the EurOtop (2018) dataset on sloped breakwaters. Among them, GPR yielded the best predictive performance, achieving an R² of 0.80 and the lowest RMSE, MAE, and RAE values (0.100, 0.013, and 0.30, respectively), indicating a strong agreement with observed data. Feature importance analysis revealed that Relative Freeboard and Freeboard Deficit (FD) were the most influential parameters across the models. To enhance interpretability and practical usability, we translated the ML findings into mathematical expressions using polynomial regression and Genetic Programming (GP). A new set of simplified equations was developed to estimate mean overtopping discharge (q) based solely on FD, effectively modelling the relationship between FD and ln(q) within the EurOtop dataset. The proposed formulae provide coastal engineers with a rapid, interpretable, and reliable tool for estimating mean wave overtopping, significantly enhancing design efficiency and decision-making under uncertainty. By bridging the gap between advanced data-driven techniques and practical engineering needs, this work advances the integration of ML into coastal infrastructure design and supports the development of more adaptive and climate-resilient defence systems.