Assessment of free vibration frequencies of nano-scale functionally graded materials using hybrid machine learning approaches

Accurate prediction of the free-vibration frequencies of nano-scale functionally graded material (FGM) beams under diverse boundary conditions is essential for the design of next-generation nanoelectromechanical systems. We develop a hybrid machine-learning surrogate in which Support Vector Regression (SVR) models the smooth global input–output trend and an Artificial Neural Network (ANN) learns the residual, yielding a compact, high-fidelity predictor. The training database is generated via finite-element analysis based on a parabolic shear deformation theory (PSDT) with nonlocal elasticity. Hyperparameters of both SVR and ANN are tuned by four recent metaheuristics, Tribe-based Intelligence Evolutionary Optimizer (TIEO), Lightning Memory Optimization (LMO), Bald Eagle Search with golden-ratio refinement (BES), and Puma Optimizer (PO), under a Monte-Carlo/SPXY protocol for robust train/validation/test splits. Performance is assessed by CC, RMSE, VAF, NSE, and the scale-independent RSR. Among all optimizers, BES consistently delivers the lowest test error (aggregate RMSE ≈ 0.176 ± 0.058, RSR ≈ 0.11 ± 0.03, NSE ≈ 0.991 ± 0.007), while PO attains the highest linear concordance (CC ≈ 0.999 ± 0.002) with a modestly higher RMSE than BES. These results indicate that the hybrid SVR-ANN, especially when tuned by BES captures ∼99 % of the explainable variance (VAF) and limits typical prediction error to ∼11 % of the observed standard deviation, offering an efficient and reliable tool for nano-FGM vibration analysis and design.