## 3 Simulation
The verification of the learned model utilized the entire system frequency according to the load change measured in the actual system as verification data. The nonlinear regression model has the advantage of expressing the complex nonlinear relationship between input and output, so it was designed as a polynomial model considering the nonlinear characteristics of the piezo element. However, this model has limitations in solving the long-term dependency problem of time series data. To compensate for this, an LSTM regression model, which is suitable for time series data processing and effective at modeling the complex dynamic characteristics of the piezo element, was additionally designed, and experiments were conducted. The simulation conditions that considered the matters for estimating the optimal resonance frequency using the above LSTM model are as shown in Table 2.  
Figure 11: Graph of the relationship between ultrasonic system resonant frequency and mechanical resonant frequency of piezo elements. The input dimension is one-dimensional data because the parameters used in the LSTM model are the ultrasonic system resonance frequency (\(f_{system}\)), the piezo element mechanical resonance frequency (\(f_{p piezo}\)), the parameter values \(L_{1}\) and \(C_{1}\) of the piezo element, and \(L_{f}\) and \(C_{f}\) of the \(LC\) filter. The hidden unit uses the six one-dimensional data mentioned above as input, and if the number of hidden units is too large or small, meaningful results cannot be obtained and the gradient vanishing phenomenon may occur. In this study, the number of hidden units was defined as 50, 100, 150, 200, 250, and 300, respectively, and the experiment was conducted, and the regression analysis was performed most stably when the number of hidden units was 200. In other words, this is the number of hidden units defined through the experiment based on the data learned in this study. The number of epochs was also not further trained because the learning rate did not increase after 200 or more training sessions. This may vary depending on the parameters used in the LSTM model or the results of the experiment. The number of epochs is the most optimized value for the data and experiments in this study. In other words, it may vary depending on the experimental environment (amount of data used for training, hardware status). The batch size was set to match the hardware of the system used in this study (GPU core performance and capacity) that conducted the training, and is a value determined by the experimental environment. Figure 12 is a diagram of the LSTM model used in this experiment. The LSTM model used in the experiment is a bidirectional LSTM model consisting of one input layer, one LSTM layer, one fully connected layer, and one regression output layer. Since using two or more LSTM layers can cause the gradient vanishing problem, it was configured as one layer. Figures 13 and 14 show the results of MATLAB learning simulations to estimate the relationship between the entire system frequency and the mechanical resonance frequency of the piezo element through regression analysis using a nonlinear regression model and an LSTM model. The ultrasonic system resonance frequency on the x-axis in Figure 11 can be seen to be divided into right and left sections based on 30 kHz due to nonlinear characteristics. Therefore, the right resonance frequency section was divided into 1 and the left resonance frequency section was divided into 2, and the analysis was conducted \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameters** & **Parameter Values** \\ \hline Input Dimension & \(1\) \\ Number of Hidden Units & 200 \\ Number of Epochs & 200 \\ Batch Size & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LSTM model hyperparameters. Figure 12: LSTM model used in the experiment. using the LSTM model. As a result, the characteristics of the entire mechanical resonance frequency range of the piezo element were estimated using the LSTM model.