## 2 Materials and Methods
On the other hand, the Monte Carlo approach yields quite uniform distribution of samples projected on the axes, yet there may be regions of high density of design points, and regions without any points at all. The best saturation of design points projected onto the axes is visible for the LHS designs. Moreover, the OLHS maximizes the uniformness of design points distribution over the design area, in order that no overlapping points nor unsampled areas are observed when utilizing this sampling technique. Figure 3: Schematic representation of different sampling plans: (**a**) Three-level factorial design, (**b**) Monte Carlo, (**c**) LHS, (**d**) OLHS. To overcome the excessiveness of a sampling plan, and to achieve multidimensional stratification (i.e., a situation in which projections of individual samples on the axes are spread uniformly), the Latin hypercube sampling is introduced. Generally, the LHS is generated in a random manner, and thus the uniform space-filling property may not be reached. Therefore, the optimal Latin Hypercube Sampling was chosen as a source of the sample data. The difference between LHS and OLHS is the optimization phase, in which the even spread of points is achieved using the \(\varnothing_{p}\) criterion, which is based on the maxmin distance criterion (minimum distance between any two sample points is maximized), as described in [23]:  
\[\min_{l\leq 1,j<n,j\
eq i}d\left(x_{i},x_{j}\right), \tag{4}\]  
where \(d\left(x_{i},x_{j}\right)\) is the distance measured for any two sample points \(x_{i}\) and \(x_{j}\), calculated as:  
\[d\left(x_{i},x_{j}\right)=d_{ij}=\left[\sum_{k=1}^{m}\left|x_{ik}-x_{jk}\right| ^{t}\right]^{\frac{t}{t}}, \tag{5}\]  
with \(t=1\) or \(t=2\) depending on the norm used (1 for the Manhattan norm, and 2 for the Euclidean norm). Once all distances are calculated, they are sorted in ascending order \((d_{1},d_{2},\ldots,d_{s})\) forming a distance vector. At the same time, the index list is created \((J_{1},J_{2},\ldots,J_{s})\), which consists of \(J_{j}\) elements specifying how many pairs of the sampling plan are separated by distance \(d_{j}\). Then, the optimal sampling plan is achieved if it minimizes the following criterion [24]:  
\[\varnothing_{p}=\left[\sum_{i=1}^{s}J_{i}d_{i}^{-p}\right]^{\frac{1}{p}}, \tag{6}\]  
OLHS assists in generating more evenly spread plans, which is not only better for the generality of the sampling plan, but also resolves problems with fitting the Kriging model, which will be discussed in more detail in the further part of the paper. For the training and validation data sets, separate OLHSs were created, each satisfying its own \(\varnothing_{p}\). The matrix optimization was terminated once the assumed time was reached. Two types of metamodels were used in the study, based on their ability to approximate (or interpolate) complex continuous and nonlinear functions without any prior assumptions regarding the function shape (which might be the case for polynomial models) and due to their universality. Studies utilizing similar metamodels were carried out, e.g., in [25]. The first metamodel is an artificial neural network (ANN) in the multilayer form. The foundations for this model were created in the 1940s by [PERSON] and [PERSON] [26] as a single artificial neuron (Figure 4a). Gathering multiple neurons in several layers together formed a simple neural network (Figure 4b), in which popularity started to grow significantly after the backpropagation learning algorithm was implemented [27]. Figure 4: Neural network concepts: (**a**) Simple perceptron, (**b**) multilayer perceptron network. There are many variants of ANNs, among which networks with radial basis function in the hidden layer are commonly used for metamodeling purposes [28; 29], due to their clear architecture and simpler learning process. On the other hand, multilayer perceptron networks may have a complex architecture with numerous activation functions, hidden layer numbers, and features, such as regularization [30; 31] or output normalization.