## 2 Theoretical Background
### Reduced Order Models  
Proper Orthogonal Decomposition (POD) [37; 38; 39; 40; 41; 42] is a powerful technique widely utilized for extracting crucial features from high-dimensional datasets. In scientific applications, it has proven effective in reducing computational costs by establishing low-order models that faithfully replicate the original system's behavior. One of the most effective methods used alongside POD is singular value decomposition (SVD) [41], which compresses the dataset into a small number of modes. These modes correspond to the dominant features of the system's behavior and are derived from thecovariance matrix diagonalization of the dataset. Each DP of the dataset is a snapshot. Given that M is the matrix composed of the learning data where each column represents the values of an individual snapshot, the SVD matrix can be expressed as follows:  
\[M=\left[ \begin{array}{ccc}\vdots&\ldots&\vdots\\ s_{1}&\cdots&s_{n}\\ \vdots&\cdots&\vdots\end{array} \right]=\left[ \begin{array}{ccc}\vdots&\ldots& \vdots\\ u_{1}&\cdots&u_{n}\\ \vdots&\cdots&\vdots\end{array} \right]\times\left[ \begin{array}{ccc}\sigma_{ 1}&\cdots&0\\ \cdots&\cdots&\cdots\\ 0&\cdots&\sigma_{n}\end{array} \right]\times\left[ \begin{array}{ccc}\vdots& \ldots&\vdots\\ v_{1}&\cdots&v_{n}\\ \vdots&\ldots&\vdots\end{array} \right]^{T}=U\Sigma V^{T} \tag{5}\]  
where  
* \(\Sigma\) is a diagonal matrix composed of \(\sigma_{i}\), the singular values of matrix \(M\). * \(U\) and \(V\) are matrices such that \(U\)\(U^{T}=Id\) and \(V\)\(V^{T}=Id\). * \(s_{i}\) are the learning solution vectors (learning snapshots i). The field solution \(X\) in the design space can then be approximated as a linear combination of r modes:  
\[X=\sum_{i=1}^{r}v_{i}\sigma_{i}U_{i} \tag{6}\]  
\(r\) is a critical parameter that balances accuracy and compression while excluding noise. It is worth noticing that in Equation (6), each snapshot \(s_{i}\) is approximated by spanning the retained modes using the retained components of the companion column of \(V\) resulting from the SVD as new modal coordinates \(v_{i}\) in the reduced space coordinates. Once the POD modes have been obtained, it is possible to define a ROM, which is a low-dimensional mathematical representation of the system preserving the essential features of the original problem. A static ROM is trained by adopting GARS to represent the variation of the outputs according to the inputs. Each snapshot \(s_{i}\) results from a state of the input \(x\), where the \(x\) vector is defined in the input space (it usually comprises physical input parameters, such as speed, load, and pressure, and geometrical input parameters, such as diameter, hole, and thickness) and can be represented in the reduced space by the retained components of the companion column of \(V\) resulting from the SVD. The GARS method could be used to represent the variation in each natural coordinate in the response space, i.e., each component of s is individually trained by a specific RS to capture its variation with respect to the input \(x\); however, this is impractical as the size of s could be high (up to many millions of dof). The POD compressed version of the data is used instead of the full one. In the compressed space, the coordinates are the weights of the modes \(v_{i}\); the size of the vector is usually smaller than one of the problem natural coordinates and is lower than the number of snapshots used, typically a small fraction. The GARS training happens on the weights, i.e., using the parameters \(x\) corresponding to each snapshot as scalar inputs to train the companion-retained components of \(V\), as computed for POD generation. This approach allows to map the input space \(x\) in the reduced space \(v_{i}(x)\) according to (7), where the dependence of a generic (unseen) new input state \(x\) is now represented. The full dataset for a new input state is recovered by a simple span of the modes using the new GARS-evaluated coefficients as weights for the new, unseen input.