Modeling the viscoelastic behavior of a FG nonlocal beam with deformable boundaries based on hybrid machine learning and semi-analytical approaches

This study investigates the free vibration behavior of Euler–Bernoulli beams made of viscoelastic materials using nonlocal theory. The mechanical properties of the nanobeam are functionally graded through its thickness, and the viscoelastic effects on energy damping are considered. Furthermore, micro- and nano-scale structural effects are incorporated into the model using nonlocal elasticity theory. Based on this, a semi-analytical solution method is developed to determine the natural frequencies and damping ratios of the beam under elastic boundary conditions. The effects of various parameters such as geometry, material grading, viscoelastic properties, and nonlocality on the dynamic behavior of beam are studied using this solution, and the results are compared with other studies in literature. Subsequently, a space-filling sampling technique is used to generate well-distributed samples of input parameters uniformly across an input space. The generated dataset is used to train various machine learning (ML) models such as k-nearest neighbor, decision tree regression, extreme gradient boosting, and light gradient boosting. Various hyperparameter optimization techniques including metaheuristic algorithms (particle swarm and genetic algorithms) and model-based methods (Bayesian optimization with Gaussian process and tree-structured Parzen estimator) are explored. A detailed study is conducted to identify the most efficient optimization technique with the most robust ML model. It is found that the decision tree regression incorporated into Bayesian optimization with tree-structured Parzen estimator) achieves the best performance in terms of computational cost and accuracy. This hybrid model requires only 11.64 s to train and perfectly predicts vibration frequencies with coefficient of determination (R²) of 1. The model's robustness is further validated using comprehensive statistical and graphical evaluations.