OPERA: A Unified Framework for AI-Assisted Polymer Metamaterial Design Through Operator Learning, Physics Embedding,...

Additive manufacturing has opened an extraordinary design space for polymer metamaterials, enabling microstructures whose macroscopic mechanical behavior is governed largely by geometry rather than by chemical composition. A principled design framework must solve two coupled problems: a forward problem (given a microstructure, predict effective properties) and an inverse problem (given target properties, generate a microstructure). Convolutional neural networks (CNNs) solve the forward problem accurately, but the inverse problem remains more challenging for three reasons reported in the literature: (i) many surrogates predict only a scalar proxy rather than the full second-order elastic tensor; (ii) fixed or randomly initialized inverse decoders create a distribution-shift gap between surrogate predictions and physical re-evaluation; and (iii) dataset bias toward near-solid configurations limits exploration of low-density and anisotropic designs. We present a unified framework, the Operator-Physics-Enhanced Reverse Architecture (OPERA), that addresses all three issues. First, the forward surrogate predicts the complete 3×3 plane-stress stiffness tensor Ceff in Voigt notation, with an analytical layer enforcing Cij=Cji and positive definiteness by construction, achieving R2>0.99 on the directional moduli and density and R2>0.88 on the off-diagonal coupling term C16 and the effective Poisson ratio. Second, a normalizing-flow decoder Fϕ, jointly trained with the forward surrogate, keeps inverse design on the training manifold and reduces the surrogate-PDE re-evaluation gap from more than 30% to below 6% on held-out targets. Third, a five-family dataset with uniform coverage of ρ∈[0.10,0.95] is augmented through an expected-improvement active-learning loop. We embed minimum-feature-size, connectivity, and print-direction constraints into the optimization through differentiable regularization and report agreement of R2=0.987 between predictions and tensile measurements on ten FDM-printed specimens. The framework is demonstrated on five problems (auxetic, extreme anisotropy, isotropic low-density, chiral, and hierarchical), with an average target error of 6.8%. The results are framed relative to a reproduced scalar-proxy baseline; we provide an explicit statistical uncertainty analysis, a baseline-reproduction protocol, and a discussion of the method's assumptions and numerical enforcement.