## 4 Discussion and Conclusions  
The acoustic sensor offers the advantages of non-contact measurement and easy installation. However, the acoustic array captures abundant fault characteristic information of rolling bearings and a significant amount of background noise. To address these issues, we propose embedding the multi-scale mathematical morphological operator into the convolutional neural network to create a novel mathematical morphological neural network named MMNet. Consequently, we have obtained the following conclusions:  
(1) The proposed method effectively mitigates noise interference in acoustic signals and successfully extracts fault characteristic information from rolling bearings. The algorithm demonstrates rapid convergence speed, achieving a fault classification accuracy of 98.56%. Experimental results indicate that the proposed method maintains high sensitivity and stability even under strong background noise and limited training samples. (2) Compared to ResNet, DenseNet, and LeNet-5, it is evident that the proposed method demonstrates superior t-SNE clustering efficacy with distinct separation among different fault types. Additionally, this method achieves higher accuracy rates than the other three methods. Regarding model training efficiency, the proposed method only takes 22.97 s for a 100-step iteration time, outperforming the other three methods. These comparative results further validate the exceptional accuracy of the proposed method in fault diagnosis. The noise addition experiment demonstrates the effective noise removal capability of the proposed method, as the EAVGH operator can simultaneously extract positive and negative pulses in acoustic radiation signals. Furthermore, different structural element scales of EAVGH exhibit varying feature extraction performance, further validating the exceptional filtering ability of MMNet. Consequently, it is anticipated that the proposed method will also yield favorable results under real working conditions. Future work will further investigate the migration algorithm and lifetime prediction method based on mathematical morphological networks. Additionally, it will explore research on rolling bearing fault diagnosis in the presence of multi-source data. Moreover, we can consider addressing the fault diagnosis problem of multi-source field coupling and researching transfer learning based on mathematical morphological networks. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Scale** & **Method** & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **Average** \\ \hline \multirow{4}{*}{10\%} & ResNet & 89.2 & 88.7 & 87.9 & 90.8 & 87.5 & 88.82 \\  & DenseNet & 72.2 & 74.5 & 75.6 & 74.5 & 78.5 & 75.06 \\  & LeNet-5 & 74.7 & 78.4 & 74.7 & 75.7 & 77.3 & 76.16 \\  & Proposed method & 96.1 & 95.9 & 95.0 & 95.7 & 93.8 & 95.30 \\ \hline \multirow{4}{*}{30\%} & ResNet & 90.3 & 90.6 & 89.5 & 91.2 & 89.1 & 90.14 \\  & DenseNet & 89.1 & 87.6 & 89.1 & 90.6 & 88.8 & 89.04 \\  & LeNet-5 & 80.1 & 77.6 & 74.3 & 76.6 & 72.1 & 76.14 \\  & Proposed method & 98.1 & 97.9 & 98.8 & 98.2 & 98.6 & 98.32 \\ \hline \multirow{4}{*}{50\%} & ResNet & 90.9 & 91.3 & 90.6 & 92.1 & 91.3 & 91.24 \\  & DenseNet & 95.2 & 93.7 & 93.1 & 95.5 & 94.8 & 94.46 \\ \cline{1-1}  & LeNet-5 & 86.9 & 85.2 & 83.8 & 87.1 & 80.1 & 84.62 \\ \cline{1-1}  & Proposed method & 98.3 & 98.1 & 98.2 & 98.2 & 98.4 & 98.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Analysis results under different data sets. **Author Contributions:** Conceptualization, Y.L. and S.K. ; methodology, Y.L. ; software, Y.L. ; validation, Y.L. and Y.Y. ; formal analysis, Y.L. and S.K. ; investigation, Y.L. ; resources, X.T. and X.K. ; data curation, Y.L. and X.K. ; writing--original draft preparation, Y.L. ; writing--review and editing, Y.L. ; visualization, Y.Y. ; supervision, F.S. ; project administration, F.S. and X.T. ; funding acquisition, F.S. and X.T. All authors have read and agreed to the published version of the manuscript.