## 5 Conclusions  
In this paper, we propose a method to estimate the optimal resonance frequency through regression analysis to maximize the mechanical output of the piezo element by measuring the frequency of the entire system combined with the input \(LC\) filter and the piezo element. The validity of the proposed method was verified through MATLAB simulation and ultrasonic system experiments, and the following conclusions can be summarized. 1. The mechanical resonance frequency of the piezo element in the ultrasonic system shows nonlinear characteristics. Therefore, a system design that considers the characteristics of the piezo element is necessary. This includes the theory of selecting each parameter value in the \(LC\) filter section and impedance matching to match the resonance frequency of the inverter output stage and the resonance frequency of the piezo element as similarly as possible in the ultrasonic system. 2. In order to operate the system at the mechanical resonance frequency to transmit maximum power to the piezo element, the creation of an equivalent model of the piezo element and an analysis of the resonance characteristics were performed. In addition,  
Figure 20: Data for estimation of mechanical resonant frequency of piezo elements. the system was designed by analyzing the impedance of the series \(R_{1}L_{1}C_{1}\) circuit that affects the mechanical output of the \(LC\) filter and the piezo element. Through this, the relationship between the system resonance frequency and the mechanical resonance frequency of the piezo element was analyzed by calculating the characteristic equation, calculating the characteristic root, calculating the damping period, and calculating the vibration period. 3. The LSTM model was used to utilize the mechanical resonance frequency of the piezo element for efficient operation of the ultrasonic system. As a result, a method to estimate the characteristics of the entire mechanical resonance frequency range of the piezo element using the LSTM model was proposed. The relationship was verified through the analysis of the mechanical resonance frequency of the piezo element, the entire system resonance frequency of the ultrasonic system inverter output stage, and the parameter values \(L_{f}\) and \(C_{f}\) of the \(LC\) filter, and the piezo mechanical resonance frequency was estimated through this. In addition, the LSTM model showed more accurate mechanical resonance frequency estimation results than the nonlinear regression model in the piezo element with nonlinear characteristics. 4. Since the LSTM model can learn complex temporal dependencies, including nonlinearity, it can better model the complex response of the piezoelectric element. Since the LSTM model shows strength in simultaneously learning and predicting multiple modes, it is effective even in complex systems with multiple frequency modes. The estimation of the mechanical resonance frequency of a piezoelectric element using the LSTM model can show superior performance in processing complex nonlinear systems compared to existing methods. In addition, it can enable more accurate and stable frequency estimation. From this study, it can be seen that using the LSTM model is effective in estimating the mechanical resonance frequency of the piezoelectric element, and it is thought that it will be effective to analyze other systems that require resonance frequency estimation using the LSTM model in the future. Conceptualization, J.M. and S.L. ; methodology, S.L. and B.K. ; software, J.K. and B.K. ; validation, J.K., J.M. and S.L. ; formal analysis, J.K. and B.K. ; investigation, S.L. and G.K.; data curation, J.M., S.L. and B.K. ; writing--original draft preparation, J.K., S.L. and B.K. ; writing--review and editing, J.M. and S.L. ; Funding acquisition, G.K.; Project administration, J.M. and S.L. All authors have read and agreed to the published version of the manuscript.