Figure 2
From: Global optimization of metasurface designs using statistical learning methods
A representative 2D analytical example to illustrate the different behaviors of CMA-ES and EGO methods. The problem considered here consists in minimizing an analytical function, known as the Branin function, characterized by the presence of 3 global minima, as indicated by the black arrows in (a). The evolution of CMA-ES as a function of the solver calls is provided in (b) in which the yellow points represent the objective function values at each iteration, the purple curve indicates the best value at each iteration. (c) Similar to (b) except that here the blue points represent the DOE phase NDOE = 6 (only 6 in this example). These points are used only for the initial training (blue shaded region in (b)) while the black dots represent the data generated during the optimization phase. The green line represents the best, optimized, results obtained during optimization phase, as explained in Fig. 1(b). (d–g) Evolution of the points tested by CMA-ES, as a function of the generation numbers (for each generation we simulate 6 designs). The yellow points represent all samples evaluated so far, the red points correspond to the last generation of size N = 6. This illustrates the search by progressive sampling and convergence. (h–k) Evaluation of the Gaussian process model (surrogate model) and the underlying database generated from iteration 0 to 30. Notice that the model converge to the analytical function shown in (a) after 30 iterations. The design points from the DOE phase are shown in blue, the black points represent the database progressively enriched during the optimization. Note that all minima are detected. As background, the GP model is plotted, which converges progressively towards the true cost function (a).
