## 3 Methodology
### Model Training  
In this study, we exploited three different autoencoders. The first is 1D CNN, in which the convolutional layer performs the function of a feature extractor. It employs appropriate filters or kernels to capture spatial and temporal interdependence in time-varying signals. To train the convolutional autoencoder, we built a hybrid model in which convolutional layers and BiLSTM were simultaneously used. The 1-D convolutional layer first filters the signal and removes the majority of the high-frequency noise. After that, the BiLSTM layer refines the signal features further. There is a total of 21 layers used with an overall 21,000 learnable. Figure 8 presents the architecture for the CNN network. In the subsequent phase, a pure LSTM autoencoder model was developed. This model consisted of multiple bidirectional LSTM layers alternated with ReLU activation layers. These were later followed by a fully connected layer and a regression layer. There are nine layers in the model, with 29,300 learnable parameters. In the third phase, we added the MODWT layer with a multiresolution analysis algorithm in biLSTM architecture. This study used \"Coiflets\" orthogonal wavelets with five transform levels. The complete architecture is given in Figure 9 and all parameters are presented in Table 2.  
Figure 8: One-dimensional CNN architecture. Figure 7: Waveform of few baseline signals at different exciter-sensor paths. Adam optimizer was used for training, which is suitable for nonstationary and non-linear data such as Lamb wave signals, with a learning rate of 0.001. The minibatch size was 500, with a maximum epoch between 200 and 600, depending on the computational complexity. All the training was done using MATLAB(r) deep learning toolbox while all network architecture was designed and analyzed on the Deep network designer app provided by Mathwork(r). Since 90% of baseline datasets are used for the model training, and 10% are used for validation purposes, the Table 3 represents the training and validation accuracy. As observed from the Table 3, when comparing with 1D CNN, it is evident that biLSTM performs better for the given datasets. However, adding the MODWT layer further enhanced the prediction accuracy of biLSTM and resulted in slightly lower RMSE \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Layers** & **Learnable Properties** &
\begin{tabular}{c} **Number of** \\ **Learnable** \\ \end{tabular} \\ \hline
1 & Sequence input & & \\
2 & MODWT & Weights 3 \(\times\) 30 & 60 \\
3 & Flatten & & \\
4 & biLSTM 1 (32 hidden units) &
\begin{tabular}{c} Input weights = 128 \(\times\) 5 \\ Recurrent weights = 128 \(\times\) 16 \\ Bias = 128 \(\times\) 1 \\ \end{tabular} & 2816 \\
5 & Relu 1 & & \\
6 & biLSTM 2 (64 hidden units) &
\begin{tabular}{c} Input weights = 256 \(\times\) 32 \\ Recurrent weights = 256 \(\times\) 32 \\ Bias = 256 \(\times\) 1 \\ \end{tabular} & 16,640 \\
7 & Relu 2 & & \\
8 & biLSTM 3 (32 hidden units) &
\begin{tabular}{c} Input weights = 128 \(\times\) 64 \\ Recurrent weights = 128 \(\times\) 16 \\ Bias = 128 \(\times\) 1 \\ \end{tabular} & 10,368 \\
9 & Relu 3 & & \\
10 & Fully Connected &
\begin{tabular}{c} Weights = 1 \(\times\) 32 \\ Bias = 1 \(\times\) 1 \\ \end{tabular} & 33 \\
11 & Regression Output & & \\  & Total & 29,900 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Proposed biLSTM + MOWDT layers. Figure 9: BiLSTM + MODWT layer autoencoder architecture. values. The model with biLSTM and MODWT layer can capture the signals' peak position, magnitude, and shape well. Figure 10 shows the validation mean square error for all three architectures. Since there are 77 validations signals, we only present the first 30 signals here. It can be observed that the biLSTM with MODWT layer outperforms the other deep learning scheme and generates signals with better quality. Furthermore, Figure 11 shows the MAE for all exciter-sensor paths for all given datasets, determined by the proposed method.