## 3 The Formulation of Spatio-Temporal Deep Learning Model for Ultrasonic Wave Propagation  
The LSTM is a unique RNN structure modeled for addressing vanishing gradients and learning long-range dependencies in the deep machine learning framework. The LSTM shown in Figure 2 consists of the following elements: a memory cell \(C_{t}\), which can accumulate, and the forgetting state being tackled, time step to time step. \(i_{t}\), \(f_{t}\), and \(o_{t}\) are the input, forget, and output gate, respectively. The \(i_{t}\) monitors the flow of new input into the \(C_{t}\), the responsibility of \(f_{t}\) is to remove irrelevant information from \(C_{t}\), and the work of \(o_{t}\) is  
Figure 1: (a) Type-1 datasets: finite-element models: FE models with single to multi-point sources excitation are triggered in the Y-direction, and an absorbing boundary condition is applied on all the domain edges, to avoid undesired reflection. (b) Type-2 datasets: finite-element models: FE models with single-point source excitation are triggered in the Y-direction on the top edge of the model and a traction-free boundary condition is applied on all other edges. to send information from \(C_{t}\) to hidden state \(H_{t}\). The \(H_{t}\) could retain the memory of past information from the sequence data. Since the nature of the training datasets is in the form of spatio-temporal sequence images, the LSTM approach suffers from capturing the spatial features from the 2D images. The ConvLSTM network, effectively, learns the spatial and temporal features from the sequential images because it uses the convolutional operation inside the LSTM cell to pair the temporal state with spatial information. The formulation of the ConvLSTM model is followed from [23]. \[i_{t}=\sigma(W_{xi}*X_{t}+W_{hi}*H_{t-1}+W_{ci}\circ C_{t-1}+b_{i}) \tag{1}\]  
\[f_{t}=\sigma(W_{xf}*X_{t}+W_{hf}*H_{t-1}+W_{cf}\circ C_{t-1}+b_{f}) \tag{2}\]  
\[C_{t}=f_{t}\circ C_{t-1}+i_{t}\circ Tanh(W_{xc}*X_{t}+W_{hc}*H_{t-1}+b_{c}) \tag{3}\]  
\[o_{t}=\sigma(W_{x0}*X_{t}+W_{ho}*H_{t-1}+W_{co}\circ C_{t}+b_{o}) \tag{4}\]  
\[H_{t}=o_{t}\circ Tanh(C_{t}) \tag{5}\]  
here, '\(W\)' denotes the weights, and '\(b\)' represents the network's biases. The convolutional operation is marked with '\(*\)' and '\(\circ\)', representing the Hadamard product. We have used the proposed SDL network for our spatial-temporal sequence generation problem, as shown in Figure 2, for two types of different training datasets. First, for type-1 datasets, we have modified the existing ConvLSTM algorithm to fuse the geometrical information with each hidden dimension output, which is similar to our previous study [26]. So, for the type-2 datasets, we have modified the suitable network parameters, such as hidden dimensions and kernel size, instead of modifying the network architecture. These models consist of an encoder and a decoder network, containing a series of convolutional LSTM cells stacked together, since a single ConvLSTM layer will not capture the forward and reflected wave dynamics from training datasets, due to the spatial-temporal nature. Each input data sequence is fed into each encoder ConvLSTM cell, to learn the wave propagation physics from the datasets. The ConvLSTM layer takes the batch of the 3D input tensor (length \(\times\) width \(\times\) time step) and outputs the hidden state (\(H_{t}\)). Then, the following input tensor, similarly with \(H_{t}\), is given input to the next layer of the ConvLSTM cell. Due to the convolutional operation, the model captures the required features from the datasets, during the processing into layers of ConvLSTM cells. The encoder, iteratively, processes input sequences through various ConvLSTM cells and outputs the embedding  
Figure 2: Typical spatio-temporal deep learning (SDL) model encoder–decoder architecture, for modelling wave propagation phenomena. tensor, representing wave propagation. The output of the encoder-embedded tensor is fed to the decoder network, to produce predicted wave propagation simulations.