Sandwich-type periodic structures function as structural metamaterials whose dynamic response is principally governed by the unit-cell geometry of the periodic architecture [1]. Such systems exhibit elastic band gaps, which are frequency intervals of strong wave attenuation. This allows for the manipulation of band gap width and location by tailoring the unit-cell configuration, enabling targeted vibration suppression [2]. The main objective of this study is the inverse design of targeted bandgaps, i.e., determining the proper unit-cell parameters for a user-defined frequency specification. This inverse problem is computationally demanding due to the high-dimensional design space and the strongly nonlinear dependence of dispersion characteristics on geometry, typically requiring repeated computationally costly eigenvalue analyses on periodic unit cells. To overcome these limitations, this study introduces a high-fidelity three-dimensional Spectral Element Method for the efficient and accurate analysis of bandgap behavior in sandwich composites. The proposed modeling framework is formulated based on the 3D elasticity equations and employs high-order Chebyshev polynomial expansions with Gauss–Lobatto sampling, yielding spectral accuracy with substantially reduced Degrees-of-Freedom compared to conventional Finite Element Analysis [3]. Dispersion relations are obtained by solving the unit-cell eigenvalue problem under Bloch–Floquet boundary conditions, which enforce periodicity through phase shifts across opposing cell boundaries. Building on this efficient forward modeling capability, a high-fidelity database is constructed for a parametric family of sandwich unit cells to support scalable learning and inverse design. Leveraging this dataset, we developed a physics-aware Gaussian Process Regression forward surrogate, wherein physical constraints are explicitly encoded into the kernel design. This surrogate achieves high predictive accuracy on held-out data, enabling rapid performance evaluation at a fraction of the numerical cost. Finally, we solve the inverse design problem using a conditional optimization framework defined over a feasible, geometry-parameterized design space, thereby automating the design of elastic metamaterials tailored to target bandgaps.