# Multi-Objective Optimization of Cascade Blade Profile Based on Reinforcement Learning
## 4 Surrogate Model Based on ANN  
The optimization, especially with multiple objectives, are computationally intensive and time-costly. Using a surrogate model to approximate and replace high-fidelity CFD simulations is necessary for the complicated optimization of aerodynamic shapes. Surrogate models can provide aerodynamic information rapidly, which reduces the computation during optimization and shortens optimization period [52; 53]. It is attractive for complex multi-objective optimizations for turbomachineries and is used in blade shape optimizations [54; 55]. ANN has excellent nonlinear mapping and learning ability, which makes it advantageous in approximating the strong nonlinear relationship bewteen the geometric shape and multiple simulated aerodynamic parameters [34; 56]. Using ANN as the surrogate model, with prediction instead of computation, eliminates the need for CFD simulations during optimization, which improves optimization efficiency and automation [56].  
In this paper, a surrogate model based on ANN is used to predict the total pressure loss coefficients, outlet flow angles, and laminar flow areas of the deformed blade profiles. One separate neural network is used for prediction of each aerodynamic parameter. During the training process of the DDPG, the surrogate model acts as the role of environment, giving the predicted aerodynamic parameters and calculating the reward of the given action and state.  
The structure of the neural network used in this paper is shown in Figure 12. The input variables are the geometrical parameters, and the output variable is the aerodynamic parameter. The network has a six-layer structure with four hidden layers. The numbers of nodes of the four hidden layers are \(60,100,80,40\), respectively. The selection of the ANN structure, including the number of hidden layers and the number of nodes in each hidden layer, is experience-based [57]. In this paper, referring to Reference [34], the structure is defined by adjusting and trying many times. The activation function of each layer is ReLU (Rectified Linear Unit) function \(f(x)\), in which the expression can be written as: \(f(x)=\max(0,x)\). The data is divided into a training set (1000) and test set (200). The optimizer of the network is [PERSON] [58]. The learning rate is set as 0.03 to make sure the network converges fast.  
Figure 11: Computation results at different number of grid points.  
Since the transition location and the total pressure loss vary consistently in a small range near the design point when the modification is reasonable and small, while the outlet flow angles vary less consistently over the operating range, the total pressure loss coefficient and the laminar flow areas at the operating point and multiple on-design and off-design outlet flow angles are considered in the optimization. One network with the above structure and hyperparameters is built and trained for each output variable in aerodynamic parameters (\(\varpi\), \(\bar{A}_{SS}\), \(\bar{A}_{PS}\), \(\bar{g}_{2}\) of \(\beta_{1}\) in \((44.66^{\circ},45.66^{\circ},46.66^{\circ},47.66^{\circ},48.66^{\circ},49.66^{ \circ})\)). The trained networks can represent the correspondence between geometric data and aerodynamic data. The aerodynamic parameters of one specific deformed rotor can be predicted easily by feeding the geometric parameters to the trained networks.