## 3 Results
### Optimization by NSGA-II  
NSGA-II was applied to optimize the rotor to improve the pressure ratio and efficiency. The initial population \(P_{0}\) was randomly generated in the normalized design space and then evolved by NSGA-II. The performance of individuals was evaluated using surrogate models in the RL environment.  
The evolved population is shown in Figure 11, and the maximum fitness of each generation was monitored as shown in Figure 11. As the evolution process continued and the calling times of surrogate models increased, the maximum fitness increased and fluctuated within a specific range. Surrogate models markedly reduced the calculation cost so that the population could evolve adequately. The population's fitness after 500 generations was plotted in Figure 11, revealing a prominent Pareto front. The evolution of each condition was repeated five more times and plotted together, verifying that the influence of random parameters in the GA process had little influence on the results. The term \(\delta f_{\rm W}\) changed the distribution of the fitness function, so the Pareto front was shorter than before, and its value changed after constraining the working point.  
Figure 11: Evolution of the rotors using NSGA-II. (**a**) Convergence history. (**b**) Pareto front.  
Figure 10: Operating characteristic variation in the rotors after different steps (from previous work). (**a**) Flow-pressure ratio. (**b**) Flow efficiency.  
Seven individuals in the Pareto front were selected to check the operating characteristics, as also marked in Figure 11b, from which the individuals with specific fitness and minimum geometry modification were selected. For the L2 normalized condition, the ranges were \(f_{1}\in[0.05,\ 0.07]\) and \(f_{1}\in[0,\ 0.01]\), while for the L2 normalized + working point condition, the ranges were \(f_{1}\in[0.04,\ 0.05]\) and \(f_{1}\in[0,\ 0.01]\).  
Figure 12 shows that the operating characteristics, pressure ratio and efficiency of the selected individuals improved. The efficiency of cases 1 and 2 improved more, and their pressure ratio improved less, while cases 3 and 4 gained a greater pressure ratio improvement, meeting the predictions of the fitness functions. Therefore, the NSGA-II algorithm worked well with the assistance of surrogate models, and the results were physically verifiable.  
The average mass flow variation at the working point of the final population \(d\overline{m}_{W}\) was calculated as \(d\overline{m}_{W}=0.0016\) before applying the working point constraint and decreased by 61.25% to 0.0069 after \(\delta f_{W}\) was added. As also shown in Figure 12b, the working points of cases 5, 6, and 7 were closer to the original case and to each other, showing that \(\delta f_{W}\) worked well.  
GA optimization can be considered one step in the design process and integrated into the RL framework. The design variables can be modified directly and continuously using the trained agent to match the expected performance. For the GA, the Pareto front has no straightforward relations to other points in the design space. The nearby points in the Pareto front can have entirely different design variables. Therefore, another evolution process was necessarily performed to generate modification steps similar to those of the agents. In addition, prior knowledge was still required to define the fitness function and select individuals using GA optimization.