Artificial intelligence has revolutionized the process of solving engineering problems by facilitating predictions and optimizations with ease. The use of artificial neural networks (ANNs) for predicting the mechanical behavior of small-scale structures, particularly in the instability field, is widely prevalent. The present study aims to use an ANN to predict the frequency response of microscale spinning cylindrical laminate shells. Our ANN design utilizes a novel modified Adam optimizer for dimensional consistency, and a genetic algorithm is used for optimizing the number of nodes in the hidden layers of the ANN. ANNs require a valid dataset for training and testing, thus, a numerical solution of the size-dependent elasticity theory is generated, and a parametric study is conducted to find the dependency of vibrational responses on different parameters of the cylindrical shell resting on an elastic foundation equipped with a piezoelectric layer. These numerical results are utilized to train and test the designed ANN. The unique feature of the current ANN is the utilization of several well-known optimizers to observe their efficacy in solving the vibrational problem. The results of the vibration study from both numerical and ANN are presented using transverse displacement of the structure. The study finds that the AdaDelta optimizer performs satisfactorily in comparison to other optimization algorithms in terms of both total error and convergence rate.