## 3 Problem Description
### Transformed Problem  
The time-series signal-processing ability of CNNs is identical to that of recurrent neural networks (such as long short-term memory) but at a faster training speed [33], and these networks have been used to process vibration-signal data in several studies [32]. In this study, a metro TGD-identification CNN (MTGDI-CNN) based on DARTS is proposed to solve the CVD-based metro-TGD identification problem. Essentially, the basic problem is transformed into an MTGDI-CNN development problem. The research framework is shown in Figure 4. This study builds a metro TGD identification model based on a CNN, and uses the CVD and heterogeneity factors collected on in-service trains to construct samples suitable for the CNN model input. At the same time, it is assumed that the detection results obtained by the TGC are reliable, and the defect detection results are considered to be actual defects on the track. Thus, the samples are labeled with the TGC-defect reports collected on the same track at the same time as the CVD, to form the CVD-TGD dataset. Based on the dataset, the DARTS method is introduced to construct a MTGDI-CNN.  
Figure 4: Framework of this study.  
Figure 3: (**A**) CVD coordinate system; (**B**) CVD acquisition location; (**C**) structure of the portable detector.  
The training process of a CNN model represents an optimization problem, as shown in Equations (3) and (4).  
\[\min_{\hat{Y}=f(X|\mathbf{w})}L\big{(}\hat{Y},Y\big{)}, \tag{3}\] \[\text{s.t.}(X,Y)\in\mathbf{D}_{\text{train}}, \tag{4}\]  
where \(\mathbf{X}\in\mathbf{R}^{n}\) denotes the sample, \(Y=0,1,2,\ldots\) indicates the label and \(\hat{Y}\) is its prediction, \(L\) denotes the loss function, \(f\) represents a CNN model, \(\mathbf{w}\) symbolizes the weights in the model, \(\mathbf{D}_{\text{train}}\subset\mathbf{D}=\{(\mathbf{X},Y)|\mathbf{X}\in\mathbf{R}^{n},Y=0,1,\ldots\}\) indicates the training dataset split from the original dataset \(\mathbf{D}\), and \(\mathbf{D}\) denotes a dataset comprising samples and labels. Each pair \((\mathbf{X},Y)\) in \(\mathbf{D}\) is termed an example, and \(\mathbf{D}\) is also named an example dataset. Further work is required to solve the MTGDI-CNN development problem.  
(1) **Design rules for CVD-TGD example dataset generation** investigates the method of transforming the original data and prediction target in the CVD-based metro-TGD identification problem into input and output suitable for MTGDI-CNN. Moreover, it needs to consider 1) the method for transforming continuous CVD into discrete samples suitable for CNN input and 2) the heterogeneous factors between CVD and TGD that need to be introduced when generating the sample.  
(2) **MTGDI-CNN architecture design, training, and evaluation**. The model architecture is designed according to the example dataset. CVD and TGD have a complex functional relationship that warrants an increase in the number of CNN layers to fit. However, adding layers to a CNN model with an ordinary simple architecture will lead to overfitting, thereby reducing the identification performance of the model [14]. Therefore, the NAS problem of MTGDI-CNN is a key problem to be addressed in the MTGDI-CNN development. This scheme must also consider (1) the reduction in the search space in Equations (1) and (2) the improvement of the training speed in Equations (2) and (3) the strategy to cope with the class-imbalance problem in the TGD dataset, and (4) the evaluation of the comprehensive performance of different model architectures in the context of the practical needs of metro management.