## 4 Results and Discussion
### SDL Model Implementation on Type-1 Datasets: Forward Wave Propagation Simulation
#### 4.1.2 Ultrasonic Wave Propagation on Different Geometrical Domain Modelling  
This section presents the comparative study between the SDL model and FE, to simulate the forward wave propagation simulations for the different geometrical domains, shown in Figure 4. So, we have considered two scenarios of different geometrical domains (i.e., other than the trained domains) with point source excitation to evaluate the trained SDL model, which is not used during training or testing. The SDL model is, initially, trained with a single-step short-term prediction during the training process. Then, the model takes a 5-input image sequence from the training set and predicts the next following consecutive 10-output image sequence. These model-predicted images are fed back to the model's input, iteratively, to predict the complete long-term simulation. Then, the SDL model generated simulation is compared with the actual FE ultrasonic wave propagation simulation. Figure 4 shows the FE and SDL model simulations over the total simulation time for geometry-1 and geometry-2. In particular, the top row in Figure 4a shows the FE datasets, and the bottom row shows the SDL model output datasets at identical time intervals at \(t\) = (0.9 \(\upmu\)s, 1.5 \(\upmu\)s, 2.2 \(\upmu\)s, 2.2 \(\upmu\)s, 3.6 \(\upmu\)s), for the long-term simulation of geometry-1. Figure 3: **(a) Spatio-temporal deep learning (SDL) model, for forward wave propagation: training and testing average loss values over the number of epochs for the type-1 datasets. (b) Spatio-temporal deep learning (SDL) model, for reflected wave propagation: training and testing average loss values over the number of epochs for the type-2 datasets. **  
Similarly, Figure 4 depicts the wave propagation of geometry-2. The proposed SDL model could capture the constructive and destructive interference at the wavefront interaction and match the FE simulation. We can conclude that the SDL model-generated simulations are in good agreement with FE. For quantitative assessment, we have extracted the displacement values at x = 17.5 mm and y = 0 to 35 mm, in all the frames from Figure 4 (as shown by the vertically dotted yellow line). Figure 5,b show the wavefront line scans for both geometry-1 and geometry-2. The line scans from the SDL model follow the same trend as the FE simulation line scans for all the frames, but some differences exist in amplitude and TOF. To find out the difference in amplitude and time of flight between the SDL and FE simulations, from the line scans in Figure 5 for each modelling scenario, we have used three methods (refer to Section 4.2) (1) the Mean Absolute Error (MAE), (2) Root Mean Square Error (RMSE), and (3) Mean Absolute Percentage Error (MAPE), to estimate the error in each frame. Table 1 shows the summary of the MAE, RMSE, and MAPE on amplitude and TOF error. The maximum MAE is in order of \(10^{-2}\), the maximum RMSE is in order of \(10^{-1}\), and MAPE is 115.9% on amplitude error, over all the time frames. The maximum MAE is in order of \(10^{-1}\), the maximum RMSE is 3.15, and MAPE is 3.34% on TOF error, over all the time frames. The evaluation finding shows that the proposed SDL model could, efficiently, simulate ultrasound wave propagation on the different geometrical domains, other than the trained domain, with good accuracy. Figure 4: Forward wave propagation of different geometrical domain: The FE and SDL are modeled using 5 MHz central frequency and two cycles with a single-point excitation source on different geometrical domains. The SDL model predicted simulations are qualitatively compared with FE simulation. (**a**) Forward wave propagation simulation for geometry-1 and (**b**) forward wave propagation simulation for geometry-2, at exact time steps for different time instances.