## 2 Random Finite Element Method
### Simulation of Non-Stationary Soil Random Field  
The random field theory extensively delineates the spatial variability of seabed soils [37; 38]. Undrained shear strength (\(S_{u}\)) constitutes an essential property of seabed soil, significantly influencing the bearing capacity of offshore foundations in undrained conditions [39; 40]. This study examines the spatially variable soil undrained shear strength to investigate the bearing capacity of skirted foundations. Several effective methods for simulating soil property random fields include the spectral representation method (SRM) [16], local average subdivision (LAS) method [41], and Karhunen-Loeve (KL) expansion technique [42; 43]. The SRM, characterized by its high efficiency and accuracy in previous research [44], is adopted to simulate the random field of \(S_{u}\).  
Numerous studies illustrate that the mean value of \(S_{u}\) increases linearly with depth for the seabed soil [45; 46; 47], which can be expressed by Equation (1):  
\[\mu_{S_{u}}=\mu_{S_{u}\cup 0}+kz \tag{1}\]  
where \(z\) is the depth of the seabed soil, \(\mu_{S_{u}}\) is the mean value of \(S_{u}\) at the depth \(z\); \(\mu_{S_{u}\cup 0}\) is the mean undrained shear strength at the mudline; and \(k\) is the increasing gradient of strength below the mudline. According to [PERSON] et al. [16], \(\mu_{S_{u}\cup 0}\) is set to 10 kPa and \(k\) remains fixed at 1 kPa/m, which indicates that the mean undrained shear strength of the soil in the numerical model increases from 10 to 60 kPa as depth ranges from 0 to 50 m.  
Determining the scale of fluctuations (SOFs) in horizontal and vertical directions is essential for evaluating the correlation degree between any two points in the random field. When the SOF approaches zero, data from any two points in the soil domain become independent. [PERSON] and [PERSON] [48] report mean values of horizontal and vertical SOFs (i.e., \(\delta_{h}\) and \(\delta_{v}\)) for submarine soil as 50.7 m and 3.8 m, respectively. The coefficient of variation (COV), which governs the dispersion degree of \(S_{u}\) in the random field around its mean value, constitutes another important parameter. Generally, [PERSON] and [PERSON] [48] suggests that the COV of submarine soil ranges between 0 and 0.5. In addition, a lognormal distribution is employed to simulate the variability of \(S_{u}\) to preclude negative values in the random field. The two-dimensional exponential autocorrelation function [49] describes the spatial correlation of \(S_{u}\), as expressed in Equation (2).  
\[\rho(\tau_{h},\tau_{v})=\exp\biggl{[}\biggl{(}-\frac{2|\tau_{h}|}{\delta_{h}}- \frac{2|\tau_{v}|}{\delta_{v}}\biggr{)}\biggr{]} \tag{2}\]  
where \(\rho\) is the correlation coefficient; \(\tau_{h}\) and \(\tau_{v}\) are the distances between any two points in horizontal and vertical directions, respectively.  
This paper aims to investigate the predictive performance of the proposed machine-learning model regarding the bearing capacity of the skirted foundations in spatially variable soil. The variation of COV, \(\delta_{h}\), and \(\delta_{v}\) are considered separately to provide a comprehensive depiction of the surrogate model's prediction performance. Table 1 lists the specific parameters involved in simulating the random field, and the detailed procedures for simulating the non-stationary random field of \(S_{u}\) are summarized by [PERSON] et al. [16]. A typical series of random field realization is presented in Figure 1.