Adaptive continuous-discrete variables optimization for active learning with extremely sparse data in optical material design

Optimizing planar multilayer (PML) optical coatings remains challenging due to the vast parametric space and complex figure of merit (FOM) requirements. This study introduces an adaptive thickness approach combined with active learning (i.e., an adaptive scheme) for concurrent material selection and thickness optimization, where thickness is adaptively sampled in a continuous spectrum, and material status is labeled as a discrete binary variable for flexible design exploration. In the adaptive scheme, we examine the performance of three machine learning (ML) models - Gaussian process regression (GPR), factorization machines (FM), and field-aware factorization machines (FFM) - for a surrogate function, and ML model-specific optimization algorithms such as discrete particle swarm optimization (PSO), artificial bee colony optimization (ABC), and simulated annealing (SA). The optimal PML structure's secondary criteria (e.g., total thickness, number of layers) are investigated and compared with the conventional fixed thickness approach (i.e., fixed scheme). As a benchmarking study, we optimize an ultrathin Ge-YF₃ antireflective PML coating on a high-index Si substrate using the adaptive scheme. It identified an optimal five-layer design with 0.47% average reflectance, requiring only ~10% of the training data of the fixed scheme and 0.002% of total possible states, reducing computational costs and enhancing practical applicability. Furthermore, we confirmed the applicability of the adaptive scheme to extended design problems, including two-dimensional photonic structures and multilayer coatings composed of four materials.