## 2 Theory and Method
### Deep Convolutional Neural Network  
In Figure 1, the architecture of the two-stage deep convolutional neural network composed by two stages, stage 1 employed as field reconstruction model and stage 2 employed as performance prediction model, is described in detail. In stage 1, the field reconstruction model with deconvolutional neural network is trained by minimum the loss function \(\ell_{stage1}\) between real and predicted fields. With the predicted fields from stage 1 as input, the performance prediction model with convolutional neural network is trained by minimum the loss function \(\ell_{stage2}\) between real and predicted performance. It should be emphasized that the \(\ell_{stage2}\) backward propagate through both the convolutional and deconvolutional neural networks if reconstructed fields are the input, while the \(\ell_{stage2}\) just backpropagate through the convolutional neural network with real fields as input. In stage 1, the deconvolutional neural network is employed to establish the reconstruction mapping from design variables. The input is design parameters \(\mathbf{x}\), while the temperature and pressure fields are target physical fields. Assuming the reconstruction mapping in stage 1 can be defined as followed:  
\[\mathbf{\hat{f}}=\hat{\mathbb{F}}_{1}(\mathbf{x};\Theta_{1}) \tag{10}\]  
where \(\mathbf{x}\) is the input design variables, \(\mathbf{\hat{f}}\) is the reconstructed field, \(\Theta_{1}\) is the learnable parameters in reconstruction deconvolutional neural network, the reconstruction mapping \(\hat{\mathbb{F}}_{1}\) can be obtained by minimizing the expectation of loss function \(\ell_{stagel}\) in the definition domain of the dataset. The training process of stage 1 is presented as:  
\[\Theta_{1}=\operatorname*{arg\,min}_{\Theta_{1}}\left[\mathbb{E}_{[\mathbf{x},\mathbf{f}]\sim\mathcal{D}}\big{(}\ell_{stagel}\big{)}\right] \tag{11}\]  
where \(\{\mathbf{x},\mathbf{f}\}\)\(\sim\)\(\mathcal{D}\) indicates design variables and fields samples obtained by numerical simulations in definition domain \(\mathcal{D}\). At stage 2, the performance is predicted from physical fields using a deep convolutional neural network. In this study, the input at stage 2 is the reconstructed fields obtained at stage 1 and the output is the interested performance of S-CO\({}_{2}\) turbine, power, and efficiency. The mapping function from physical fields too performance can be described as follows:  
\[\hat{\psi}=\hat{\mathbb{F}}_{2}\big{(}\hat{\mathbb{t}};\Theta_{2}\big{)}=\hat{ \mathbb{F}}_{2}\big{(}\hat{\mathbb{F}}_{1}(\mathbf{x};\Theta_{1});\Theta_{2} \big{)} \tag{12}\]  
where \(\hat{\psi}\) is the predicted turbine performance, \(\hat{\mathbf{f}}\) is the reconstruction field at stage 1, \(\Theta_{2}\) is the learnable parameters of the deep convolutional neural network, the mapping function \(\hat{\mathbb{F}}_{2}\) can be obtained by minimizing the expectation of loss function \(\ell_{stagel2}\) in the definition domain of the dataset. The training process of stage 2 is formalized as:  
\[\Theta_{2}=\operatorname*{arg\,min}_{\Theta_{2}}\left[\mathbb{E}_{[\mathbf{f},\mathbf{\psi}]\sim\mathcal{D}}\big{(}\ell_{stagel2}\big{)}\right] \tag{13}\]  
where \(\{\mathbf{f},\mathbf{\psi}\}\)\(\sim\)\(\mathcal{D}\) indicates fields and performance samples obtained by numerical simulations in definition domain \(\mathcal{D}\). It is obvious that the input design variables of stage 1 are low dimensional data while the physical fields with high dimension are obtained as output. Thus, deconvolutional neural network is utilized to expand low-dimensional input to high-dimensional fields. Deconvolutional neural network was first proposed by [PERSON] [28] and the general application was presented in their following works [29; 30]. With the development of deconvolutional neural network, plenty of applications are conducted on scene segmentation [31], image processing [32], and so on.