## 3 Methodology
### POD-MCNN Model
#### 3.3.2 Surrogate Modeling with Multiscale Convolutional Neural Networks  
The multiscale convolutional neural network model (MCNN) is designed to accomplish the task of mapping input parameters \(C^{(i)}\) to basic mode coefficient matrices \(Y^{(i)}\). The same feature map will have different expression forms under two different scale spaces. The single-scale convolution process uses the same scale convolution kernel to ex-tract features from the input. It is easy to ignore other precision features, resulting in in-complete feature expression information. In contrast, the multiscale convolutional neural network can use multiple convolution kernels to perform convolution at different scales, with the input to extract features of different scales. Then, these features are fused to achieve the purpose of complementary information. The structure of the network containing three parallel multiscale convolutional layers is shown in Figure 5.  
The model input \(C=(c_{1},c_{2},\dots,c_{n})\) is the parameter obtained from the physical entity sensors through the MQTT protocol. In order to eliminate the influence of the different dimensions between the parameters and speed up the network gradient descent to find the optimal solution, \(C\) is standardized. After that, the input parameters are preprocessed through the fully connected neural network \(C^{\prime}=f(C)\), and the low-dimensional input signal is increased in dimension, which is beneficial to extracting the multiscale features of the input signal through the multiscale convolution operation. The features extracted by convolution kernels at different scales are different. Convolutions with larger kernel sizes have larger receptive fields but are not sensitive to detailed features. Convolutions with smaller kernel sizes can capture detailed features more effectively. Three multiscale convolutional layers are designed in our model, and each layer contains three kernel scales of \(1\times 1\), \(3\times 1\), and \(5\times 1\). These convolutional kernels are used for the multiscale feature extraction of the upper layer output. In addition to the size \(F\) of the kernel, the convolutional output is also affected by the kernel step size \(S\), the number of channels \(n\), and the padding size \(P\). The padding operation is to add zeros around  
Figure 5: Multiscale convolution. the boundaries of the input matrix. The size of the output matrix after the convolution operation can be calculated as:  
\[N_{l+1}=\frac{(N_{l}-F+2P)}{S}+1 \tag{6}\]  
where \(N_{l}\) is the size of the lth layer feature map and \(N_{l+1}\) is the size of the \(l+1\) th layer feature map. To ensure the high nonlinearity of the proposed model, we use a multi-layer deep structure for our neural network [39]. In addition, the model also fuses the original features with multiscale features through skip connections as the input of the next layer. The features extracted by the multiscale convolutional layer are finally outputted through the fully connected layer. Figure 6 shows the structure of MCNN and Table 1 lists the main parameters of the network. The activation function enhances the nonlinear expression ability of the network. LeakyRelu is chosen as the activation function:  
\[LeakyRelu(x)=\left\{ \begin{aligned} & x,x>0\\ & ax,x\leq 0\end{aligned} \right. \tag{7}\]  
where \(a\) is taken as 0.01. LeakyRelu addresses the possible neuron death problem in Relu. It has a small positive slope in the negative region so that it can perform backpropagation even for negative input values, and solved the problem that the gradient of the Relu activation function disappears. Figure 6: The framework of MCNN (“+” represents “concat” operation). In general, convolutional neural networks have a pooling layer after convolution. The pooling layer in CNN can reduce the dimensionality of features to some extent, but this is not a necessary operation for CNN [40]. In this experiment, there are not many features after convolution. There is no need to reduce the dimensionality.