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

TL;DR

This work introduces a POD–DeepONet framework to jointly address forward and inverse band-structure design in 2D photonic crystals with binary, pixel-based $p4m$-symmetric unit cells. By projecting the high-dimensional Bloch-map onto a fixed POD trunk and learning reduced coefficients with a neural branch, the authors create a differentiable, low-cost forward surrogate, enabling gradient-based inverse design for dispersion-to-structure and band-gap targets. They prove continuity and a universal-approximation property for the surrogate and demonstrate substantial gains in accuracy and robustness over baseline methods across forward predictions and two inverse tasks. The approach significantly accelerates high-contrast photonic design workflows and provides a principled path toward reliable, data-driven inverse engineering in metamaterials. Potential impact includes faster materials screening, reliable gap engineering, and a foundation for uncertainty-aware inverse design in photonic-crystal devices.

Abstract

We 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-to-structure 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 gradient-based 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.

Figures (9)

• Figure 1: Illustration of a square-lattice unit cell $\Omega$ (left) and the corresponding first Brillouin zone $\mathcal{B}_F$ (right). In $\Omega$, blue denotes alumina with permittivity $8.9$ and white denotes air with permittivity $1$; in $\mathcal{B}_F$, the IBZ $\mathcal{B}$ is the shaded triangle with vertices $\Gamma=(0,0)$, $X = (\pi/a,0)$, and $M = (\pi/a,\pi/a)$.
• Figure 2: Example of a unit cell and its band structure: (a) the $16\times16$ unit cell; (b) the first $10$ TE band functions along the high--symmetry path $\mathcal{K}_{\rm hs}$.
• Figure 3: Pixel-based parametrization of a unit cell with $p4m$ plane symmetry. Blue pixels represent the high-permittivity material with dielectric constant $\epsilon_{\rm alum}$; white pixels represent the low-permittivity background $\epsilon_{\rm air}$. The red triangle marks the fundamental wedge. Its intersection with the pixel grid yields $N_f=36$ wedge pixels $\{P_j^{\rm w}\}_{j=1}^{N_f}$, each controlled by a binary design variable $\rho_j\in\{0,1\}$. Applying the $p4m$ rotations and reflections indicated by the coloured arrows maps these wedge pixels to the full set of pixels in the unit cell.
• Figure 4: Schematic workflow of the POD--DeepONet framework. The top panel summarizes forward evaluation: snapshot POD constructs a fixed trunk, the branch network is trained using a band MSE loss on standardized data (Algorithm \ref{['alg:POD-DO-full']}), and the resulting surrogate predicts band structures for query designs. The bottom panel shows the two POD--DeepONet-based inverse design procedures, where dispersion-to-structure and band-gap targets are handled by gradient-based inverse training with data, binarity, and proximity terms (Algorithm \ref{['alg:inverse-design']}).
• Figure 5: Representative $16\times16$ pixel-based unit cells from the data set. Blue pixels denote the high-permittivity material $\epsilon_{\rm alum}$ and white pixels the background $\epsilon_{\rm air}$. The red staircase region in each panel marks the design wedge $\boldsymbol{\rho}$ with $N_f=36$ independent pixels used to parametrize the unit cells.

Theorems & Definitions (12)

• Theorem 2.1
• Remark 2.2: Generality of the pixel-based parametrization
• Remark 2.3: Relaxed optimization and modeling perspective
• Proposition 3.1: POD truncation error on the snapshot set
• Remark 3.1: fixed $\mathbf k$-grid in the POD trunk
• Proposition 3.2: Continuity of the discrete band map
• proof
• Theorem 3.2: Approximation properties of POD--DeepONet
• proof
• Remark 3.3