## 3 Data Analysis Methodology
### Dataset Bifurcation Strategy, Training of Baseline DNNs, and Transductive Analysis  
The first step of training any DNN is to create a training and testing split of the available data from both \(K_{t}\)s (i.e., \(K_{t1}\) and \(K_{t2}\)), as shown in Figure 10a. Although fairly common in all machine learning analyses, the distribution of this split is often ad-hoc (80-20% in this case) and is based on an intelligent estimate of the problem. The reason to explicitly mention this step is to emphasize the subsequent parametric analysis that delves into understanding the effects of such data splits. At this stage, however, following the train-test split, a separate DNN (Figure 10b) is trained and tested for each \(K_{t}\) to create a baseline for further performance comparison. The DNN is represented using a consistent nomenclature \(DNN_{Train:\,XY}^{Test:\,YY}\) where \(XX\) and \(YY\) represent the training and testing data used for the analysis, respectively. For example, the baseline DNN for \(K_{t2}\) (denoted as \(DNN_{Train:\,20K_{t2}}^{Test:\,20K_{t2}}\)) is trained using 80% of the available data and tested on the remaining 20%. The DNN has a fully connected structure with seven dense hidden layers and one dense output layer with a single neuron. The network receives raw, unprocessed signal data as its input. The hidden layers use the ReLU (Rectified Linear Unit) activation function [35],  
Figure 8: _‘Healthy’_ and _‘cracked’_ labeling of the ultrasonic signals for (**a**) \(K_{t1}\) and (**b**) \(K_{t2}\).  
Figure 9: (**a**) Using the _healthy_ and _cracked_ labels from the confocal microscope, an individual DNN can be trained and tested for either \(K_{t1}\) or \(K_{t2}\) using the respective dataset. (**b**) A single mixed DNN can be trained and tested from both \(K_{t1}\) and \(K_{t2}\) data.  
while the output layer uses a sigmoid activation function [35]. The model is inspired by the encoder-decoder architecture [35], where inputs are compressed to a 2D latent space and then expanded again to reconstruct the original input. A low-dimensional latent space created due to such a structure can help in interpreting the behavior of the DNN in future studies. Logistic regression [35] is carried out on the reconstructed output. Since the task is a mutually exclusive binary classification problem, binary cross-entropy [35] is used as the loss function.  
Figure 10: (**a**) Training and testing dataset bifurcation independently for \(K_{t1}\) and \(K_{t2}\). (**b**) Baseline DNN training and testing with respective datasets for both \(K_{t}\)s. (**c**) Baseline DNN architecture. (**d**) Transductive analysis—evaluating pre-trained DNNs in (**b**) by switching the test datasets for each \(K_{t}\).  
The hyperparameters for the model, i.e., the number of neurons for each hidden layer and the learning rate, are selected through a grid-search algorithm using KerasTuner [36] to ensure the optimality in terms of accuracy and speed of convergence. The optimum network (Figure 10b) has 428, 132, 96, 2, 96, 132, and 428 neurons in the seven layers, with a learning rate of 0.0004 for the Adam optimizer [36]. The fully-connected DNN architecture is shown in Figure 10c. The vast volume of data ensures that the computation rarely encounters over-fitting, and hence techniques such as \(L_{2}\) regularization and dropouts have not been used in this model. Following the construction of these baseline DNNs, their pre-trained capabilities are tested on data from another \(K_{t}\) without any training. \(DNN_{Train:\ 80K_{t2}}^{Test:\ 20K_{t2}}\) and \(DNN_{Train:\ 80K_{t2}}^{Test:\ 20K_{t2}}\), therefore, evaluates the universal applicability of pretrained DNNs (\(DNN_{Train:\ 80K_{t1}}\) and \(DNN_{Train:\ 80K_{t2}}\)) across different stress concentration factors, as shown in Figure 10d. Such an analysis is also termed as 'transductive' analysis in the machine learning literature.