## 3 Simulation
Figure 13a shows the estimation of the mechanical resonance frequency of the piezo element using a nonlinear regression model for Section 1, and Figure 13b shows the estimation of the mechanical resonance frequency of the piezo element using the LSTM model, respectively. Figure 14a shows the estimation of the mechanical resonance frequency of the piezo element using the nonlinear regression model for Section 2, and Figure 14b shows the estimation of the mechanical resonance frequency of the piezo element using the LSTM model, respectively. Figure 14a shows the estimation of the mechanical resonance frequency of the piezo element using the LSTM model, respectively. In order to improve the learning accuracy of the two models, the min-max normalization of the data set was applied as a preprocessing step. The configuration of LSTM was set to the number of input dimensions 1, the number of hidden units 200, the number of epochs 200, and the batch size 100, as shown in Table 2. As a result of analyzing the frequency relationship, the RMSE (Root Mean Square Error) of the nonlinear regression model was 0.1922 and the RMSE of the LSTM model was 0.0124. It was confirmed that the LSTM model showed a significantly lower RMSE value than the nonlinear regression model, providing more precise results for estimating the resonance frequency of the piezo element. This is interpreted as because the LSTM model effectively captures the long-term dependency of time series data and can better model the complex nonlinear behavior of the piezo element. In this study, a nonlinear regression analysis technique was applied to model the nonlinear characteristics of the piezo element. Nonlinear regression analysis is a statistical method that models the complex nonlinear relationship between input variables and output variables and can overcome the limitations of linear regression and capture more complex data patterns [34]. The general form of a nonlinear regression model is as follows:  
Figure 14: Estimation of mechanical resonant frequency of piezo elements using nonlinear regression model and LSTM model 2. Figure 13: Estimation of mechanical resonant frequency of piezo elements using nonlinear regression model and LSTM model 1. \[y=f(x,\beta)+\epsilon \tag{29}\]  
Here, \(y\) is the dependent variable, \(x\) is the independent variable, \(\beta\) is the model parameter, \(f\) is the nonlinear function, and \(\epsilon\) is the error term. In this study, a polynomial regression model was used. Polynomial regression is a form of nonlinear regression that models complex relationships using multiple orders of input variables [35]. Specifically, a ninth-order polynomial model was applied, which has the following form:  
\[y=\beta_{0}+\beta_{1}x+\beta_{2}x^{2}+\beta_{3}x^{3}+\beta_{4}x^{4}+\beta_{5} x^{5}+\beta_{6}x^{6}+\beta_{7}x^{7}+\beta_{8}x^{8}+\beta_{9}x^{9}+\epsilon \tag{30}\]  
This model attempted to capture the complex nonlinear characteristics of the piezo element by including terms from \(x\) to \(x^{9}\). The least squares method was used to estimate the parameters of the nonlinear regression model. This method finds parameters that minimize the sum of squares of the residuals [36]. The RMSE was used to evaluate the model's fit and generalization ability, and the model's performance was verified through cross-validation. The polynomial regression model has the advantage of modeling complex nonlinear relationships, but there is a risk of overfitting due to the use of high-order terms [37]. Considering this, the balance between model complexity and fit was evaluated using information criteria (Akaike information criterion, Bayesian information criterion). The nonlinear regression model was effective at capturing the complex characteristics of the piezo element but it had limitations in dealing with the long-term dependency of time series data. To complement this limitation, the LSTM model was additionally applied.