## 1 Introduction
This hybrid approach, which combines the established robustness of PID control with the innovative adaptability of deep learning, represents a significant advancement in the field of vibration control for cantilever beams. There have been many studies on the use of PZIs to control the vibration of structures. Studies have investigated the use of PZIs to control the vibration of cantilever beams subjected to a variety of excitations, including random excitation, impact excitation, and parametric excitation [4]. These studies have shown that PZIs can be effective in suppressing vibrations under a wide range of conditions. In addition to the location of the PZIs, the type of controller used also affects the effectiveness of vibration control. Several different controllers have been used in studies of PZT-based vibration control, including linear controllers, nonlinear controllers, and intelligent controllers. In one of the recent studies, the analytical vibration models of a damped cantilevered beam under a PID controller have been discussed. The beam damping coefficient and integration of the voltage limit for the PZT actuator were considered by the model. According to analytical results, the PID controller effectively suppresses vibration, and implementing the actuator voltage limit results in a rapid reduction in vibration [5]. A piezoelectric layered smart plate was used to test the developed control techniques under varied conditions. Based on the first-order shear deformation (FOSD) hypothesis, an electro-mechanically coupled finite element (FE) model of smart structures was built to design a suitable controller for the vibration suppression of thin-walled smart structures when considering the vibrations caused by various disturbances, which include both free and forced vibrations, and a PID control was utilized to dampen both the forced and free vibrations [6; 7]. In another study, a fuzzy logic controller was used for a smart cantilever beam for active vibration control. The smart cantilever beam experiences an impact force at the free end of the beam, which causes it to vibrate. The piezoelectric sensor layer generates a sensor voltage as a result of these disturbances, which the fuzzy logic controller uses as an input. Using 49 if-then rules, a fuzzy logic controller determined the actuator voltage. When this voltage is delivered to the piezoelectric actuator layer, the smart structure responds by producing a control force. Three diverse types of piezoelectric materials were utilized for active vibration control [8; 9]. Also, an adaptive controller has been applied to the vibration control in many structures. For example, an adaptive robust sliding mode vibration control (SRVC) scheme for a flexible beam used a piezoceramic sensor and actuator. The controller is designed to be robust to uncertainties in the system parameters and external disturbances. The controller can adjust its parameters automatically in response to changes in the system. The implementation of the controller on a flexible beam with a piezoceramic sensor and actuator results in reducing the vibration. The experiments show that the SRVC controller can effectively suppress the vibration of the beam under the influence of uncertainties and disturbances [10]. The main contribution of this paper is to explore and compare two deep learning-based approaches for vibration control in cantilever beams. 1. The first approach involves the direct application of deep learning techniques, specifically multi-layer neural networks and RNNs, to control the beam's dynamic behavior. 2. The second approach, which is not reported yet in the literature, integrates deep learning into the tuning process of a PID controller, optimizing its parameters for improved control performance. To activate the structure, two different input signals are used, an impulse signal at time zero and a random one.