## 2 Theory and Method
### Deep Convolutional Neural Network
The deconvolutional operation is illustrated in Figure 2 with a simple example with padding size b = 1, stride size s = 2, and kernel size k = 3. For a more convenient description, the input, kernel, and output are marked in blue, gray, and green, respectively. The input of size \(3\times 3\) is interpolated with zero and the size of intermediate matrix up to \((\mathbf{s}\times 3+\mathbf{b})\times(\mathbf{s}\times 3+\mathbf{b})\), that is \(7\times 7\). The final output is the result of convolutional operation between the kernel and intermediate matrix with stride of 1. In this point, deconvolution can be seen as a kind of special convolution. Convolutional neural networks became more and more popular in computer vision [33; 34], nature language [35], and so on due to their powerful ability of feature extracting and learning. It is a natural idea to utilize convolutional neural networks to extract the low-dimensional performance from the high-dimensional physical fields. In mathematics, the convolution operation is a kind of multiplication of input and kernel at certain strides, as shown in Figure 3. Similar to Figure 2, the input, kernel, and output in this convolutional example are marked in blue, gray, and green, respectively. As shown in the sketch, the intermediate matrix can be obtained by input with padding around the original input matrix. Additionally, then the elements of output are the multiplication results of kernel and corresponding input elements moving at a specified stride. With the complexity of application scenarios, the convolutional neural network goes deeper and deeper. However, some obstacles such as degradation of training accuracy and vanishing/exploding gradients arise with deeper layers. In order to avoid the above problems, some outstanding means, data normalization, intermediate normalization layers [36], and Residual Neural Network (ResNet) [37], were applied in our approach. In this study, input and output were normalized to \((-1,1)\) by maximum and minimum normalization, and batch normalization was applied after every deconvolution or convolution except for the output layer. The detailed architecture of our two-stage algorithm is listed in Table 1, in which Deconv2d means deconvolutional operation, Conv2d means convolutional operation, k is the size of kernel, s is the stride size, c is the channel size of output, in is the input size of linear layer, and out is the output size of linear layer. For more convenient description, the building block in ResNet is separated to the basic block (a pair of \(3\times 3\) filters) and shortcut (connection operation with identify) in this study. As shown in Table 1, the input of the field reconstruction model is firstly reshaped to a feature of large size by linear layer for subsequent deconvolutional operations. The size of output features become a specified \(256\times 64\times 4\) after the transformation of six deconvolutional layers and then the output features are interpolated to \(256\times 64\times 4\) and \(256\times 64\times 4\) for physical fields of stator and rotor blades. The similar interpolation operation can be found in performance prediction model. The physical fields with different sizes are adjusted to the specified size of \(256\times 64\) by an interpolation operation which makes the performance prediction model away from the affection of input size. After the subsequent convolutional operations from layer 1 to layer 6, the average pooling [38] and linear layers are adopted to obtain objective output from extracted features. In addition, the active function ReLU [39] is employed to enhance the nonlinear performance of the deep convolutional neural networks. Figure 3: A simple sketch of the convolutional layer. Figure 2: A simple sketch of the deconvolutional layer. The Adaptive Moment Estimation (Adam) [40] optimizer was adopted in the optimization process. In essence, it is Root Mean Square Prop (RMSProp) [41] with a momentum factor.