Supplementary material for "Hybrid Analytical–Neural Network Modelling for Damped Vibrations of Size-Dependent Functionally Graded Nanoplates on Viscoelastic Foundations"

This study presents a hybrid modelling approach for the damped vibration analysis of functionally graded nanoplates supported by generalized viscoelastic foundations. The plates, composed of ceramic–metal mixtures, are described by higher-order shear deformation theory and a newly formulated modified nonlocal strain gradient theory, accounting simultaneously for nonlocal and strain gradient effects. The foundation model extends the visco-Pasternak type to include two stiffness parameters and two damping coefficients. Closed-form Navier solutions are derived and used both for a comprehensive parametric study and as training data for a neural network surrogate model. The surrogate enables rapid vibration predictions without repeating the full analytical process. The results demonstrate strong consistency between the analytical and ANN-based predictions, confirming the reliability of the proposed hybrid approach. Parametric results highlight the coupled influence of size-dependent effects, gradation profiles, and foundation damping on dynamic characteristics. The proposed framework effectively balances theoretical rigor and computational efficiency, providing a practical reference for vibration prediction and preliminary design of micro- and nano-scale structural components.