# Multi-Objective Optimization of Cascade Blade Profile Based on Reinforcement Learning
## 6 Results and Discussion  
The convergence curves of the neural networks are shown in Figure 14. The aerodynamic parameters, except \(\widetilde{A}_{SS}\), have a good convergence result with MSEs (mean square errors) by less than \(10^{-5}\). \(\widetilde{A}_{SS}\) varies much for profiles in the profile library; thus, it does not converge as well as other parameters, but it is still within small MSE of \(10^{-4}\). The small convergence errors show that the ANNs have good performance of prediction on the test sets. The independence study of division of train-set and test-set is shown in Appendix A.2. Figure 15 shows the reward curve of the DDPG algorithm and the objective function curve of the optimization. The DDPG network can be trained well and obtain a high reward in 100 episodes. With the trained DDPG network, the optimizer can find and record a profile with very good performance in no more than 200 episodes, which, as it turns out, means that this optimization method based on DDPG can rapidly achieve a fairly good result. The \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \(\overline{\omega}\) & \(\widetilde{A}_{SS}\) & \(\widetilde{A}_{PS}\) & \(\beta_{21}\) & \(\beta_{22}\) & \(\beta_{23}\) & \(\beta_{24}\) & \(\beta_{25}\) & \(\beta_{26}\) \\ \hline
1 st PC & \(-34\times 10^{-16}\) & \(7.6\times 10^{-2}\) & \(-1.3\times 10^{-1}\) & \(5.6\times 10^{-1}\) & \(-5.3\times 10^{-2}\) & \(1.5\times 10^{-1}\) & \(2.8\times 10^{-1}\) & \(3.8\times 10^{-1}\) & \(-1.6\times 10^{-1}\) \\
2 nd PC & \(-4.1\times 10^{-1}\) & \(5.7\times 10^{-1}\) & \(3.9\times 10^{-2}\) & \(-6.9\times 10^{-2}\) & \(-1.2\times 10^{-1}\) & \(-2.4\times 10^{-1}\) & \(-3.0\times 10^{-1}\) & \(-3.0\times 10^{-2}\) & \(6.8\times 10^{-2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The 1 st and 2 nd principal component of objective space analysis. better-performance profile updates slow and varies slightly after 200 episodes, as well as the objective function. With the ANN-based surrogate model and the DDPG based optimization, a new rotor blade profile with lower total pressure loss coefficient and larger laminar flow area is achieved. By CFD validation, the performance characteristics and the flow field characteristics of the cascade with the optimized blade profile can be known and compared with the original cascade. Figure 16 shows the original and the optimized rotor blade profiles. Compared to the initial profile, the optimized profile has following characteristics: (1) The thickness of the front of the profile is slightly reduced; (2) the thickness of the middle part of the profile is increased, where the thickness at the pressure surface is increased slightly resulting in a flatter shape, and the thickness and curvature at the suction surface is obviously increased. Figure 14: Convergence curves of ANNs. Figure 15: The accumulative reward and objective function loss of each episode. The aerodynamic performance curves of the cascades of the original blade profile and the optimized profile are shown in Figure 17, respectively. In the operating region, with \(\beta_{1}\) being \(45.16^{\circ}\) to \(48.66^{\circ}\), \(\overline{\omega}\) of the cascade with the optimized blade profile decreases  
Figure 16: Initial and optimized blade profiles: comparison in terms of shape. considerably, with a relative reduction up to 6.07%. The variation of \(\beta_{2}\) is small, and the relative change in the operating region is no more than 0.31%. The performance characteristics of the cascades at the operating point are listed in Table 5. \(\overline{\omega}\) of the cascade goes from 0.01642 to 0.01583, with a relative reduction of 3.59%. The relative laminar flow area at the suction surface \(\widetilde{A}_{SS}\) rises from 34.56% to 43.35%, with massive increase. The relative laminar flow area at the pressure surface is basically unchanged.