A POD–DeepONet Framework for Forward and Inverse Design of 2D Photonic Crystals

A B S T R A C TWe develop a reduced-order operator-learning framework for forward and inverse band-structure design of two-dimensional photonic crystals with binary, pixel-based p4m-symmetric unit cells. We construct a POD–DeepONet surrogate for the discrete band map along the standard high-symmetry path by coupling a POD trunk extracted from high-fidelity finite-element band snapshots with a neural branch network that predicts reduced coefficients. This architecture yields a compact and differentiable forward model that is tailored to the underlying Bloch eigenvalue discretization. We establish continuity of the discrete band map on the relaxed design space and prove a uniform approximation property of the POD–DeepONet surrogate, leading to a natural decomposition of the total surrogate error into POD truncation and network approximation contributions. Building on this forward surrogate, we formulate two end-to-end neural inverse design procedures, namely dispersion-tostructure and band-gap inverse design, with training objectives that combine data misfit, binarity promotion, and supervised regularization to address the intrinsic non-uniqueness of the inverse mapping and to enable stable gradientbased optimization in the relaxed space. Our numerical results show that the proposed framework achieves accurate forward predictions and produces effective inverse designs on practical high-contrast, pixel-based photonic layouts.