## 6 Conclusions  
This study successfully integrates time-frequency domain transformation techniques with the residual VGG-based Siamese network to perform across-load fault diagnosis of induction motors with limited data conditions. The feature extraction layer of the model effectively converts paired time-frequency images into low-dimensional feature vectors. Additionally, multiple similarity measurement methods have been introduced to accurately gauge the distances between feature vectors, thereby enhancing image classification accuracies. Utilizing few-shot learning techniques, the model achieves exceptional diagnostic results, even under highly imbalanced data conditions, demonstrating significant practicality and efficiency. This effectively addresses the prevalent issue of training data imbalance in deep learning applications. During experimental comparisons, the method proposed in this study was benchmarked against commonly used transfer learning models like ResNet50, and deep learning models such as DCGAN and CNN. Faced with health-to-fault imbalance ratios of 20:1 and 10:1, the proposed method exhibited the highest diagnostic performance, achieving an average accuracy rate of over 98% and the lowest standard deviation compared to other models, showcasing superior stable predictive classification results. Moreover, even as the data imbalance ratio increased from 10:1 to 20:1, the diagnostic performance of the method remained robust. The learning rate adjustment strategy is identified as a sensitive parameter in this study, influencing the model's learning efficiency and ultimate classification performance. Initially, a higher learning rate was employed during the model  
Figure 10: Imbalance ratio of 20:1-100% to 50% load. (**a**) Residual VGG-based Siamese network: five-shot five-way. (**b**) Residual VGG-based Siamese network: one-shot five-way; (**c**) Fine-tuned ResNet50. (**d**) DCGAN and CNN [23]. Figure 9: Imbalance ratio of 20:1-50% to 100% load. (**a**) Residual VGG-based Siamese network: five-shot five-way. (**b**) Residual VGG-based Siamese network: one-shot five-way. (**c**) Fine-tuned ResNet50. (**d**) DCGAN and CNN [23]. training, which was then halved every 20 epochs. This strategy aids in rapidly approaching optimization early in training and allows for the fine-tuning of model parameters later to avoid excessive parameter oscillations and enhance convergence stability. Based on experimental adjustments and previous research experiences, future work could explore more learning rate adjustment strategies to further enhance the model's adaptability and performance in more complex or variable datasets. From the experimental results, the method proposed in this study makes significant contributions to the across-load fault diagnosis of induction motors with limited data conditions. This includes technical integration, addressing limitations, tackling data imbalance, and facilitating cross-domain fault diagnosis. The study has achieved notable results but also identifies limitations and suggests future research directions. Currently, the model primarily diagnoses specific types of induction motor faults. Future work could extend to more motors and other mechanical equipment to test the model's generalizability. Furthermore, considering the complexity of real industrial environments, further verification of the model's stability and reliability under various environmental conditions should be conducted. Conceptualization, H.-C.C. and C.-C.K. ; methodology, R.-G.L. ; validation, H.-C.C., R.-G.L., and C.-C.L. ; formal analysis, H.-C.C. and R.-G.L. ; investigation, C.-C.L. ; resources, C.-C.L. ; data curation, H.-C.C. ; writing--original draft preparation, R.-G.L. ; writing--review and editing, C.-C.L. and C.-C.K. ; supervision, H.-C.C. and C.-C.K. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable.