## 3 Proposed Leak Detection Model  
Using the data preprocessing methods for feature extraction described in Section 2, the 1D RMS and frequency-domain data were converted and utilized for training the proposed model for fine leak detection. The proposed model comprises two stages, as shown in  
Figure 3: Fusion of 1D pattern features using four sensors for model training. Figure 2: Features of RMS and frequency patterns for leak and noise signals: (**a**) RMS pattern feature, (**b**) frequency pattern features, where the spectrum size and pattern of the leak signal, generated through BPF filtering, are clearly distinguishable from those of the noise signal. In Stage 1, a 1D CNN structure is utilized to extract local feature maps from the preprocessed 1D RMS and frequency data, focusing on identifying fine leak or noise signals. The 1D convolution operation is mathematically defined in Equation (6):  
\[ \begin{array}{l}x_{(d,n)}^{l}=f_{l}\Big{(}\sum_{m\in R_{k}^{l}}\Big{[}x_{(d,m) }^{(l-1)}\times w_{(d,(m,n))}^{l}\Big{]}+b_{(d,n)}^{l}\Big{)},\\ d\in\{T,F\},n=1,2,\ldots,N_{d}^{l},\ l=1,2,\ldots,\ L_{d},\end{array}  \tag{6}\]  
where \(d\) denotes the domain, with \(T\) denoting the time domain and \(F\) the frequency domain. \(F.x_{(d,n)}^{l}\) denotes the 1D feature map \(n\) of domain \(d\) at layer \(l\), with dimensions \(1\times W_{d}^{l}\). \(x_{(d,m)}^{(l-1)}\) denotes the 1D feature map \(m\) of domain \(d\) at layer \((l-1)\), whereas \(w_{(d,(m,n))}^{l}\) denotes the kernel function with a window size of \(1\times C_{d}^{l}\), mapping from feature map \(m\) of domain \(d\) at layer \((l-1)\) to feature map \(n\) at layer \(l\). The bias term is denoted by \(b_{(d,n)}^{l}\), and \(f_{l}(\cdot)\) denotes the activation function. \(N_{d}^{l}\) denotes the number of output feature maps at layer \(l\), \(L_{d}\) denotes the total number of convolution layers in domain \(d\), and \(R_{n}^{l}\) denotes the list of 1D feature maps at layer \((l-1)\) connected to feature map \(n\). To pre-extract features for leak and noise signals, a 1D convolution layer with a kernel size of \(1\times 3\) and eight filters is applied, as summarized in Table 2. This convolution operation is followed by batch normalization (BN) and MaxPooling1D, which compress the extracted features. This process corresponds to the 1D CNN block (1) within the 1D CNN module, forming the initial feature extraction phase for fine leak detection. Next, a second 1D convolution layer with the same kernel size and 16 filters is applied, followed by BN and global max-pooling to extract the main representative features. This \begin{table}
\begin{tabular}{c c c} \hline \hline
**Stage** & **Output Dimension** & **Layer** \\ \hline Input & \(40\times 4\) & Input Layer \\ \hline
1D CNN block (1) & \(38\times 8\) & \([1\times 3,8]\) \\ \hline
1D CNN block (2) & \(17\times 16\) & \([1\times 3,16]\) \\ \hline \multirow{3}{*}{Classifier} & \multirow{3}{*}{1} & Global Max Pooling \\  & & FC: \([8\to 1]\) \\ \cline{1-1}  & & sigmoid \\ \hline \# params & & 745 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Detailed structure of single 1D CNN model for both time and frequency domains. Figure 4: Architecture of a feature fusion DL model based on stepwise TL using an attention mechanism. corresponds to the 1D CNN block (2) within the same module. The 1D convolution employs the rectified linear unit (ReLU) as its activation function. To enable independent learning for each domain, the final output feature map \(x_{d}\) of domain \(d\) is used as the input for a binary classifier. The first layer of this classifier contains eight neurons activated by a ReLU function, while the output layer includes a single neuron with a sigmoid activation function.