## 2 Theory and Method
### Deep Convolutional Neural Network
By combining the advantages of RMSProp and Adaptive Gradient (AdaGrad) [42], the Adam has lower calculation cost. In addition, it has good performance for high-dimensional space, large data sets, and most nonconvex optimization. Mathematically, the definitions of Adam are as follows:  
\[t\gets t-1 \tag{14}\]  
\[g_{t}\leftarrow\
abla_{\theta}\ell_{t}(\theta_{t-1}) \tag{15}\]  
\[m_{t}\leftarrow\beta_{1}\cdot m_{t-1}+(1-\beta_{1})\cdot g_{t} \tag{16}\]  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]0  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]1  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]2  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Layers** & **Stage 1: Field Reconstruction Model** & **Stage 2: Performance Prediction Model** \\ \cline{2-3}  & **Basic Block** & **Shortcut** & **Basic Block** & **Shortcut** \\ \hline Input size & Batch Size \(\times\) 5 (number of design variables) & \
\begin{tabular}{c} Stator blade: Batch Size (64) \(\times\) grid size (\(36\times 104\)) \\ Rotor blade: Batch Size (64) \(\times\) grid size (\(51\times 232\)) \\ Interpolation (Batch Size \(\times\) 256 \(\times\) 64 \(\times\) 4) \\ Conv2d (k = 3, s = 1, c = 32) \\ Conv2d \\ Layer 1 &
\begin{tabular}{c} (k = 3, s = 2, c = 512 \\ Deconv2d \\ (k = 3, s = 1, c = 512) \\ Deconv2d \\ (k = 3, s = 2, c = 256) \\ Deconv2d \\ (k = 3, s = 1, c = 256) \\ Deconv2d \\ (k = 3, s = 1, c = 256) \\ Deconv2d \\ (k = 3, s = 2, c = 128) \\ Deconv2d \\ Deconv2d \\ (k = 3, s = 2, c = 128) \\ Dec  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]3  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]4  
where subscript \(t\) indicates the iteration step of the optimization process, \(\ell\) is the loss function which can be \(\ell_{stage1}\) or \(\ell_{stage2}\), \(\theta\) is the learnable parameters of the neural network (\(\theta\in\Theta_{1}\) or \(\theta\in\Theta_{2}\)), \(m\) is the first moment estimation, \(v\) is the second moment estimation, \(\beta_{1}\) and \(\beta_{2}\) are attenuation coefficients, \(\alpha\) is the learning rate, and \(\varepsilon\) is a small number which prevents division by zero in the implementation. In this study, \(\varepsilon=10^{-8}\), \(\beta_{1}=0.9\), and \(\beta_{2}=0.999\). The optimization in stage 1 is performed such that the defined loss is minimized. Firstly, the field loss in this study is the mean square error (MSE) between the predicated field \(\hat{\mathbf{f}}\) and the original field \(\mathbf{f}\), which can be written as:  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]5  
where \(N\) is the sample size and \(C\) is the number of fields in the train dataset. The subscript \(i\) represents sample index and the superscript \(l\) is the type of the field. To circumvent a very blurred predication only by the field loss, the absolute error of gradient information between the predicated field and ground truth is reckon in loss function, also called gradient loss, that is defined as:  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]6  
Then the total loss function to be minimized can be written as a combination of field loss and gradient loss, where \(\lambda\) is the loss weight. In this study, \(\lambda=0.1\).