## 4 Results and Discussions
### Comparison of Wasserstein and Sequence-Based Operators  
#### 4.2.1 Wasserstein-Based vs. Reference Evolutionary Algorithm  
In this Section, the default Wasserstein-based algorithm, WBGEA_1t, is compared to the baseline evolutionary algorithm, Ref_alg (Table 1 for the acronyms).  
We observe that WBGEA_1t works better than Ref_alg in terms of both convergence speed and final value on the wind-farm problem (Figure 13). Ref_alg, with its classical crossover and Gaussian mutation, is not efficient at learning designs with specific geometrical characteristics such as alignments, which are important in the case of wind farms. The left plot in Figure 14 shows that WBGEA_1t converges faster than Ref_alg on \(F_{minDist}\).  
As is often seen in the experiments, the behavior of the algorithms is different with the inertia function, \(F_{inert}\). There, as seen on the left plot of Figure 15, Ref_alg outperforms WBGEA_1t. To explain this, another version of WBGEA_1t is tried where the probability to apply the boundary mutation, prob, is raised to 1, which simultaneously prohibits any full domain mutation. The boundary mutation is, of course, adapted to \(F_{inert}\) since its optimum has points only on the boundary. WBGEA_1t with only boundary mutations (prob = 1) yields similar final \(F_{inert}\) values as Ref_alg, although it converges more slowly (left plot of the Figure 15).  
Figure 11: Alternating vs. Successive Boundary and Full Domain Mutations. The test function is the inertia function, \(F_{inert}\). (**Left**): mean \(\pm\) std. deviation of the best so far solution. (**Right**): mean \(\pm\) std. deviation of the population diversity. Statistics calculated from 20 independent runs.  
Figure 12: Alternating vs. Successive Boundary and Full Domain Mutations. The test function is the MinDist function, \(F_{minDist}\). (**Left**): mean \(\pm\) std. deviation of the best so far solution. (**Right**): mean \(\pm\) std. deviation of the population diversity. Statistics calculated from 20 independent runs.  
The diversity of WBGEA_1t decreases to zero more rapidly on the wind farm proxy (right plot of the Figure 13) and \(F_{minDist}\) (right plot of Figure 14) than on \(F_{inert}\), where the drop is slower (right plot of Figure 15). For these situations, the loss of diversity is associated with a convergence towards a global optimum. For the same reason, increasing the rate of application of the boundary mutation by changing prob from 0.5 to 1 helps in locating global optima and decreases diversity.  
Concerning Ref_alg, the diversities of the populations stagnate at a high level for all the test functions (see right plot in Figures 13-15). While a convergence of the population to a global optimum induces, by definition, a loss of diversity, the experiment with Ref_alg on function \(F_{inert}\) shows that evolutionary algorithms have other long term behaviors: the population remains very diverse, yet the best points approach the optimum consistently throughout repeated runs. The preservation of a high diversity is thought to be an effect of the constant Gaussian mutation strength, \(\sigma\), and of the crossover over sequences, which exchange points from the crossed sets.