## 3 Mathematical Model
### Surrogate Modeling
#### 3.1.2 Polynomial Chaos Expansion  
A polynomial chaos expansion (PCE) is a way of constructing an analytical model that maps the outputs of interest to inputs using a predefined basis of polynomials. The  
Figure 2: **(a)** Single layer neural network [33], and (b) six-layers deep neural network used in the present study.  
function is represented in the form of a polynomial expansion [34]. In a PCE, we assume a deterministic map, M, such that the response is \(Y=\text{M}(X)\) where \(\text{x}\in\Re^{m},Y\in\Re^{n},m,\) and \(n\) represents the input and output varibles respectively For simplicity let us consider \(n\) to be \(1\). The probability distribution of \(X\) is given by the probability density function \(f_{X}(x)\)[35]. Thus, the polynomial expansion of \(Y\) can be written as the following equation:  
\[Y(X)=\sum_{\alpha=1}^{NP}a_{\alpha}\varnothing_{\alpha}(X) \tag{13}\]  
where \(a_{\alpha}\) represents the expansion coefficients. The value of \(NP\) depends on the order of polynomial \(p\) and the number of input variables \(m\). Thus, the exact value of \(NP\) can be calculated by \(NP=\frac{(m+p)!}{m!p!}\). The other term in Equation (13), \(\varnothing_{\alpha}\), represents the multivariant basis functions, which are orthogonal with respect to the probability density function \(f_{X}(x)\), that is:  
\[\int_{\Re^{m}}\varnothing_{\alpha}(x)\varnothing_{\beta}(x)f_{X}(x)dx=\delta_ {a\beta} \tag{14}\]  
Here, \(\delta_{a\beta}\) represents the Kronecker symbol. For independent input variables, the value of \(\varnothing_{\alpha}\) can be obtained by the tensor product of the univariate orthogonal polynomials \(\varnothing_{p_{i}^{\alpha}}(X)\), as in:  
\[\varnothing_{\alpha}(X)=\prod_{i=1}^{m}\varnothing_{p_{i}^{\alpha}}(X_{i})) \tag{15}\]  
where, \(p_{i}^{\alpha}\) represents the multi-index vector. The univariate polynomial basis function was selected with respect to the type of probability density function. For example, if we have a uniform distribution, the Legendre polynomial is considered the optimal basis function. The PCE coefficient \(a_{\alpha}\) can be computed using a regression approach, by minimizing the mean square error \(\sum_{j=1}^{N}\left(y_{D}^{j}-Y(x_{D}^{j})\right)^{2}+\gamma a^{T}\text{a}\). Given a dataset D of \(N\) input variables \(X_{D}=\left(x_{D}^{1},x_{D}^{2},\ldots,\ x_{D}^{N}\right)\) and their corresponding output vector \(\gamma_{D}=\left(y_{D}^{1},y_{D}^{2},\ldots,\ y_{D}^{N}\right)\), the expansion coefficients can be calculated by solving the following problem:  
\[a=\left(\theta^{T}\theta+\gamma I\right)^{-1}\theta^{T}Y_{D} \tag{16}\]  
where \(\gamma\) is the regularization factor, \(I\) is the identity matrix and \(\theta\) is the design matrix, whose components are \(\varnothing_{j}\left(x_{D}^{i}\right)(i=1,..,N;j=1,..,NP)\). The total number of sample input points are calculated using \(N=\Upsilon NP\), while \(Y\geq 1\) represents the oversampling parameters that control the accuracy of the PCE. To generate the dataset, we can use any sampling method, including Latin hypercube or Sobol [36]. Once we obtain the expansion coefficients, we can use the relationship (13) to approximate the outputs for any input variables.