## 1 Introduction  
Soft soils, such as clay, have high compressibility and low shear strength. Thus, soft soils are considered unsuitable for construction, and ground improvement is required for construction on soft soils. In recent years, aggregate piers have been extensively used to increase bearing pressure and reduce settlement and lateral displacement under structural footings. Moreover, aggregate piers act as vertical drains and accelerate the consolidation of surrounding soft clay. The prediction of the ultimate bearing capacity of the improved ground is an important task for a proper design [1]. Since the 1970s, many researchers have aimed to develop a methodology based on plasticity theory [2], co-expansion theory [3; 4], numerical methods [5; 6], and empirical methods [7; 8]. Predictive models are constantly being proposed and updated. Laboratory and field footing load tests on the aggregate pier have been conducted to investigate the behavior of the aggregate pier and its bearing capacity [9; 10; 11; 12]. Numerical analysis was used to identify stress and strain acting on the piers and surrounding soil according to the load applied to the footing, and experimental results were used to prove the numerical analysis. [PERSON] and [PERSON] [13] performed numerical analysis to simulate the behavior of a stone column, and it was verified by comparison with the experimental results. [PERSON] et al. [14] presented a numerical model to simulate the performance of a single stone column and groups of stone columnsinstalled in soft clay. [PERSON] and [PERSON] [15] studied the behavior of stone columns through a laboratory test and numerical study. [PERSON] and [PERSON] [16] performed a three-dimensional numerical modeling of an aggregate pier system considering the installation effects. In addition, [PERSON] et al. [17] presented an analytical model to predict the bearing capacity of soft soil reinforced with stone columns, and the proposed model was validated via the laboratory and numerical results. [PERSON] and [PERSON] [18] evaluated the accuracy of existing bearing capacity models using 30 footing load tests with various field conditions and noted that existing models have shown a broad range of bias, variation, and error. The prediction errors of existing models were frequently extensive and unsuitable for practical design. Therefore, these authors proposed a modified ultimate bearing capacity model, and the prediction accuracy was considerably better than existing models. However, the ultimate bearing capacity models were modified using the relationship between the undrained shear strength and bearing capacity factor (or cavity expansion factor): this relationship was derived from the load test results of a single aggregate pier with a high area replacement rate (0.9-1.0), and the prediction error of the ultimate bearing capacity was larger for groups of aggregate pier than for a single aggregate pier. They also proposed a multiple linear regression (MLR) model based on 29 load tests, and it showed favorable performance. However, there was no evaluation of the input variables selection, and the verification was conducted for only one independent load test. [PERSON] et al. [12] performed statistical analysis using the Statistical Package for the Social Sciences (SPSS), and proposed a regression equation to predict the bearing capacity. However, they also did not consider the effect of the input variables selection, and the validation with independent data was not performed. Additionally, although each input variable may have a different nonlinear relationship with the bearing capacity, both studies considered the same equation type of input variables (exponential or power). In recent years, using artificial intelligence (AI) techniques to address problems in civil engineering has amplified [19; 20; 21]: the artificial neural network (ANN) is an AI technique that is applied to estimate ultimate bearing capacity [22; 23].