## 2 Materials and Methods
### Global Sensitivity Analysis  
The machine-learning model introduced in Section 2.3 is used to model the relation of the JCA model parameters and the microscale geometry of the additively manufactured specimens--here, the design variables. In order to gain insight about the underlying physics, the machine-learning model is analyzed using global sensitivity analysis (GSA). Within this section, a summary of the mathematical descriptions of the sensitivity measures is given. In-depth information about GSA can be found in various literature; here, mainly the textbook [54] should be mentioned. All the following information on GSA is based on this reference. The sensitivity \(c_{i}\) of an arbitrary function \(g\) regarding its inputs \(\mathbf{Z}=\{z_{1},\ldots,z_{i},\ldots,z_{n}\}\) can be seen as its derivatives with regard to the inputs, i.e.,:  
\[c_{i}=\left.\frac{\partial\ g}{\partial\ z_{i}}\right|_{z_{i}}. \tag{2}\]  
This procedure is often referred to as one-factor or one-at-a-time sensitivity analysis, and is based on the first-order second moment approach [55]. The partial derivative in Equation (2) can be computed in a non-intrusive way using a parameter variation and applying finite differences. However, the procedure has (at least) two major drawbacks. Being a first-order method, the relation for the sensitivity given in Equation (2) is a linear approximation. Thus, for general (nonlinear) functions \(g\), the resulting sensitivity measure is only valid in the proximity of the nominal value of \(z_{i}\). Another drawback is that interdependencies of the input variables for not purely additive functions \(g\) cannot be accounted for. A more general and realistic sensitivity measure can be computed using the approach of the GSA. Thereby, the sensitivity is measured by the amount of variance in the output quantity that is caused by the variance of each input quantity. Since the input variance is a global measure in the sense that the amount of variance accounts for the entire input range that each \(z_{i}\) can take, the value of the output variance is also accounting for the entire input range, and thus, sensitivity measures valid for the entire parameter range are obtained. Additionally, the influence of interdependencies of the inputs can be measured to some extend. In order to evaluate the sensitivity of the model output regarding its inputs, a statistical approach is used. Following a Monte Carlo Simulation of the model, the variance of the output \(\mathbb{V}(Y)\) and the variance of the conditional expectation of the model with regard to the input \(\mathbb{V}(\mathbb{E}(Y|X_{i}))\) are measured. With these quantities, the sensitivity coefficient of the model regarding its \(i\)-th input can be computed using:  
\[S_{i}=\frac{\mathbb{V}(\mathbb{E}(Y|X_{i}))}{\mathbb{V}(Y)}. \tag{3}\]  
Figure 4: Correlations of correct model parameters (input to the ANN) and the prediction of the ANN. The data shown are the test dataset only (unseen data during training). The correlation is measured using [PERSON]’s correlation coefficient \(\rho\). Greek letters denote the JCA model parameters, i.e., \(\Xi\): flow resistivity, \(\Phi\): porosity, \(\alpha_{\infty}\): tortuosity, \(\Lambda\), \(\Lambda^{\prime}\): viscous and thermal characteristic length, respectively. The sensitivity coefficients \(S_{i}\) are referred to as first-order effects, and measure the main effect of the \(i\)-th input quantity on the model output. It should be noted that, for purely additive models, all first-order effects add up to one, i.e.,:  
\[\sum_{i}S_{i}=1. \tag{4}\]  
The first-order effects account for the direct (main) effect of the \(i\)-th input quantity on the model output. However, there can be interaction effects: for example, simultaneous changes of two input quantities may compensate or amplify each other. Such effects cannot be accounted for by a simple one-at-a-time sensitivity analysis. However, the framework of the GSA allows for the measurement of such effects using the interaction effects \(S_{ij}\).