Large-scale graph-machine-learning surrogate models for 3D-flowfield prediction in external aerodynamics

Advanced Modeling and Simulation in Engineering Sciences

Table 5 For each model, we display a metric that is based on the \(L^2\)-norm of the CpT residuals, accounting for all the test geometries

The metric appears in the form \(\mu \pm \sigma \). In particular, \( \mu = \frac{1}{10} \sum _{j=1}^{10} \frac{1}{n_j} \Vert \varepsilon \Vert _{2j}^2 \) and \( \sigma = \sqrt{\frac{1}{10} \sum _{j=1}^{10} \left( \frac{1}{n_j} \Vert \varepsilon \Vert _{2j}^2 - \mu \right) ^2}\), where \(\Vert \varepsilon \Vert _{2j} = \sqrt{\sum _{i=1}^{n_j} |\varepsilon _{ij} |^ 2}\) is the \(L^2\)-norm of the CpT residuals for the j-th geometry. \(\varepsilon _{ij}\) represents the difference between the CpT of the model and the CpT of the CFD simulation, respectively, for the i-th node of the j-th geometry of the test set; \(i=1,...,n_j\) where \(n_j\) represents the total number of nodes in the j-th geometry, while \(j=1,...,10\)