Derivative-enhanced Bayesian optimization for broad-bandgap phononic metamaterials with hypercomplex automatic differentiation

The design of Phononic Metamaterials (PM) with unique dynamic behaviors and wave propagation characteristics remains a significant challenge due to the highly non-linear relationships between design parameters and response. The arrangement of the periodic unit cells within PM is crucial for determining their dynamic behavior, making optimization methods essential for the design and development of these materials. These methods are used to tailor bandgap characteristics such as bandwidth and frequency location by optimizing the unit cell’s geometric parameters. However, existing approaches often suffer from slow convergence rates, entrapment in local minimum, or require numerous expensive evaluations of the objective function. To address these challenges, this work proposes using a novel derivative-enhanced Bayesian optimization (DEBO) framework that integrates Hypercomplex Automatic Differentiation (HYPAD) with a Gradient-Enhanced Gaussian Process (GEGP) interpolator surrogate model. This combination enables the accurate and efficient computation of objective function sensitivities, resulting in more reliable and data-efficient surrogate models. As a result, DEBO significantly improves the robustness of BO against local minima, which is particularly beneficial for the non-convex optimization problem characteristic of PM design. The framework is applied to optimize the geometry of a two-dimensional cross-shaped unit cell, maximizing bandgap width at low mid-frequencies. By consistently converging to the global optimum, we demonstrate that DEBO outperforms traditional methods, including derivative-free Bayesian optimization, gradient-based numerical optimization, and metaheuristics. Furthermore, experimental validation of the optimized geometry aligns closely with numerical predictions, confirming the effectiveness of the approach.