Gene-Expression Programming, Evolutionary Polynomial Regression, and Model Tree to Evaluate Local Scour Depth at Culvert Outlets

Protection of the downstream of culvert outlets against scour process, as a water conveyance structure, is a highly significant issue in design of culverts. Frequent field and experimental investigations were carried out to produce a relationship between the scour depth due to the governing variables. However, existing empirical equations do not always provide a precise estimation of the scour depth due to the complexity of the scour phenomena. In this investigation, gene-expression programming (GEP), model tree (MT), and evolutionary polynomial regression (EPR) are utilized to predict the scour depth downstream of culvert outlets. Input variables—considering effective parameters on the scour depth—were defined as sediment size at downstream, geometry of culvert outlets, and flow characteristics in upstream and downstream. Experimental datasets to develop the models were collected from different literature. Performances of the proposed models for the training and testing phases were assessed using several statistical measures. Results of performances indicated that EPR provided the lowest level of precision including index of agreement ($\mathrm{IOA}=0.958$) and root mean squared error ($\mathrm{RMSE}=0.419$) for prediction of local scour depth at culvert outlets than those obtained using MT ($\mathrm{IOA}=0.947$ and $\mathrm{RMSE}=0.471$) and GEP ($\mathrm{IOA}=0.943$ and $\mathrm{RMSE}=0.487$). In terms of accuracy, all proposed equations extracted from artificial intelligence approaches had remarkable superiority to the traditional equations. Ultimately, it has been proven that mathematical expressions given by evolutionary computing tools had sufficient generalization to present an accurate prediction of the local scour depth with respect to preserving physical meaning of results.