## 1 Introduction
[6] investigated three-dimensional scour rates below a pipeline exposed to currents. They proposed an empirical equation for the prediction of longitudinal scour rates below the pipeline. Additionally, [PERSON] et al. [9] studied the scouring mechanism below a pipeline in two various flow conditions (i.e., combined wave and current, wave only). They proposed two regression-based equations for the prediction of the longitudinal scour rate in both flow conditions. In terms of the propagation of scour rates around pipelines, it can be inferred that there is limited knowledge on the scour hole development. Previous attempts generally indicated that empirical equations are restricted to the range of experimental databases. Thus, these equations often do not have high potential of generalizing the prediction of scour rates in the three-dimensional space below pipelines. In the case of experimental investigations, it may be expected to find comprehensive influences of various parameters on the scouring process below pipelines. An understanding of the free span propagation still requires in-depth investigations ([PERSON] and [PERSON] [2], [PERSON] et al. [6], [PERSON] et al. [9]). With the advent of soft computing techniques, complicated mechanisms of real-life problems have become more understandable than before. In this way, there is a high demand for the presentation of robust mathematical models on the basis of Artificial Intelligent (AI) techniques in various fields of science. Over the last decade, a large number of  
Figure 3: Schematic drawing of scouring vortices due to waves at a submarine pipeline. Wake vortexes downstream of the pipeline (**a**,**b**) due to oscillatory flow. Figure 2: Schematic diagram of onset scouring process for submarine pipelines under currents. (**a**) Vortex A in front of the pipe, Vortex B in the corner downstream of the pipe, and the largest Vortex C behind the pipe; (**b**) vortex shedding that forms after the scour develops. AI approaches have been employed prosperously to obtain an accurate estimation of the local scour depth below pipelines exposed to currents and regular waves. In fact, Artificial Neural Networks (ANNs) ([PERSON] and [PERSON] [15], [PERSON] et al. [16]), Adaptive Neuro-Fuzzy Inference Systems (ANFIS) ([PERSON] et al. [17]), Support Vector Machine (SVM) ([PERSON] et al. [18]), Classification and Regression Tree (CART) ([PERSON] et al. [16], [PERSON] and [PERSON] [19]), Linear Genetic Programming (LGP), Gene-Expression Programming (GEP) ([PERSON] and [PERSON] [20], [PERSON] and [PERSON] [21]), Genetic Programming (GP) ([PERSON] and [PERSON] [22]), Group Method of Data Handling (GMDH) ([PERSON] et al. [23], [PERSON] et al. [24]), Multivariate Adaptive Regression Splines (MARS) ([PERSON] [25]), and Evolutionary Polynomial Regression (EPR) ([PERSON] and [PERSON] [21]) proved to be successful applications. In the case of scour rates prediction, there have been a few endeavors to estimate three-dimensional free spans via GMDH and ANN models below a pipeline exposed to regular waves ([PERSON] and [PERSON] [26], [PERSON] et al. [27]). However, few investigations have been conducted to estimate scour propagation rates at seabed pipelines by AI models. In fact, the experimental detection of three-dimensional scour rates is generally expensive due to the need of advanced sensors and other facilities. To the best of the authors' knowledge, powerful AI techniques such as MARS, GEP, and MT have not yet been applied in the prediction of scour propagation rates around seabed pipelines exposed to regular waves. As main merits, these AI techniques have potential in three main areas: (i) Presenting (non-linear) explicit equations for limited number of data; (ii) reducing the number of influential variables; and (iii) preserving the physical consistency of the generated models. The present paper moves in these directions, identifying new equations to predict 3D scour patterns at seabed pipelines depending on key dimensionless parameters.