## 1 Introduction
The efficacy of transfer learning is also contingent upon data distribution between the source and target domains, which can adversely affect the model's classification accuracy [30]. In addition, no standardized criterion for determining the optimal number of layers to fine-tune. If the similarity between source and target domain data is low, more layers may require fine-tuning. However, excessive fine-tuning of many layers might lead to difficulties in model convergence. Conversely, fine-tuning too few layers can degrade the model's classification performance. In summary, real-world industrial scenarios frequently encounter challenges such as obtaining fault data, the diversity of fault types, high costs associated with labeling, and specific challenges in industrial applications. In response to these issues, this study introduces the few-shot learning model based on the residual VGG-based Siamese network, designed to address deficiencies in training data effectively. This model was employed for the across-load fault diagnosis of induction motors with limited data availability. Two different data imbalance ratios were established in experimental scenarios, and various sampling methods were compared and analyzed to ascertain the superiority of the method proposed in this study. The principal contributions of this study are outlined below:  
1. Technical Integration: Combining the improved Siamese network with the time-frequency domain transformation technique can offer a holistic approach to the problem, notably improving diagnostic accuracy with limited data conditions. 2. Addressing Limitations: This paper proposed a residual VGG-based Siamese network comprising two key architectures: the feature extraction network-residual VGG and the merged similarity layer. The residual VGG architecture is designed to overcome certain shortcomings of deep neural networks, such as degradation problems and overfitting. Meanwhile, the merged similarity layer uses multiple distance metrics in a low-dimensional feature space to enhance the model's classification performance. 3. Tackling Data Imbalance: For across-load fault diagnosis in induction motors, we developed experimental models for four common types of motor failures. Vibration signals for healthy and fault conditions were measured under different load levels. Comparative analyses with other leading AI models, such as transfer learning models-ResNet50 and deep learning models integrating DCGAN with CNN [23], demonstrate the superior performance of our proposed model under data-imbalanced scenarios. Remarkably, our model achieved an average accuracy rate as high as 98% when the training dataset imbalance ratio was 20:1. 4. Cross-Domain Fault Diagnosis: Existing studies focus primarily on single, common motor-bearing faults. In contrast, our experimental setup incorporates stringent real-world industrial conditions, including data imbalances, multiple types of faults, and different load levels. Our model performed well in across-load fault diagnosis in induction motors with limited data conditions. This highlights the model's generalizability and its high applicability in fault diagnosis. In conclusion, this study successfully integrates the time-frequency domain transformation technique with the residual VGG-based Siamese network to address challenges in across-load fault diagnosis for induction motors. The organization of this paper is as follows: Section 2 describes the experimental design, including fault types, data acquisition methods, and processing procedures; Sections 3 and 4 provide a detailed introduction to the theoretical foundations of the residual VGG-based Siamese network model and the time-frequency domain transformation technique; Section 5 presents the experimental results and offers a comparative analysis of the performance of the proposed method with other advanced models; Section 6 summarizes the entire paper, emphasizing the main contributions of this research.