## 2 Proposed Multi-Fidelity Method  
Table 1 enumerates all the steps included in the comprehensive method designed in this work, which provides multi-fidelity surrogate models for design space approximation. The table distinguishes between the actions required in the HF and MF frameworks. The mathematical notation featured in Table 1 avoids particularities of the problem used as an example (condensation in mixing streams) and, thus, allows researchers from other fields to use the proposed method.  
The first stage of Table 1 is the _Conceptual modeling_, in which one defines the complete dataset \(X_{N}=\left\{x_{i}^{l}\right\}\) by identifying the model parameters \(i\) and determining the particular values for each parameter at level \(l\). Section 5.1.1 shows an example of such steps. The conceptual modeling ends by selecting a set of intermediate variables \((z_{m})\) that will be assessed for their use as surrogates, together with the target variable, which is the one that is required to be mapped \((z_{t})\). For the studied application, these variables are defined in Section 5.1.2.  
The next section of Table 1 is the _Identification of significant parameters_, which aims to reduce the computational effort of the surrogate construction. In this way, a subset of \(X_{scr}\) points can be determined using a certain design of experiments (a fractional factorial design is suggested in this work; see Section 3.1.1) and then run High-Fidelity models \(\gamma_{scr}^{HF}\) and Low-Fidelity models \(\gamma_{scr}^{LF}\) at these working points to obtain a multi-fidelity evaluation. After the screening analysis over all metamodeled variables \(z\) (using, for instance, ANOVA, as described in Section 3.2), the non-significant parameters can be removed, leaving only \(N^{\prime}\) significant parameters for the subsequent steps.  
The last stage of the proposed method (see Table 1) is the _Surrogate modeling (for space design exploration)_ devoted to the surrogate modeling itself. Firstly, \(s\) surrogate candidates are selected (see Section 3.3), and a set of training points \(X_{tra}\) is determined using another design of experiments. Here, the Box-Behnken design is employed, as justified in Section 3.1.2. Next, High-Fidelity \(\big{(}g_{s,t,}g_{s,m}^{HF}\big{)}\) and Multi-Fidelity \(\big{(}g_{s,m}^{MF}\big{)}\) surrogate models are constructed using the evaluation of High-Fidelity \(\gamma_{tra}^{HF}\) and Low-Fidelity models \(\gamma_{tra}^{HF}\) at the training points. These surrogate models are then used to predict values of the target variable at the verification points \(\hat{Y}_{s,t,}\), \(\hat{\gamma}_{s,m,per}^{HF}\) and \(\hat{Y}_{s,m,per}^{MF}\). Notice that these works consider the screening subset \(X_{scr}\) to be used as verification subset \(X_{ver}\) to reduce the computational cost. Since the exact results are already available from stage 2.2 of Table 1, the error metrics (as defined in Section 3.4) are evaluated, allowing the different combinations of HF or MF surrogate models with the target or intermediate variables to be classified as adequate or not in terms of accuracy. Finally, the best couple of surrogate \(s\) and variable \(z\) is selected to be used for space design exploration at any working point within the assessed range \(X_{exp}\).