<b>Supplementary material for "Neural network-based and analytical solutions for damped vibration investigation of the functionally graded nanoplates on viscoelastic foundations</b><b>"</b>

This study develops a hybrid computational framework combining analytical modeling and artificial neural network (ANN)-based prediction for damped vibration analysis of functionally graded nanoplates resting on viscoelastic foundations. This foundation model is an extension of the visco-Pasternak foundation with two stiffness parameters and two damping coefficients. The nanoplates, composed of a mixture of ceramic and metal, are modeled using higher-order shear deformation theory and modified nonlocal strain gradient theory. This theory accounts simultaneously for nonlocal and strain gradient effects. A closed-form analytical solution based on Navier’s method is derived from the exact solutions, while the ANN is used as a fast and efficient tool for approximate predictions. Extensive simulations are conducted to investigate the influence of foundation parameters, material gradation, geometric configurations, and small-scale coefficients on the dynamic behavior of the nanoplates. The proposed methodology demonstrates an effective integration of physics-based modeling with data-driven techniques, offering a computationally efficient tool for simulation-based analysis in nano-engineering applications.