Symbolic Learning of Topological Bands in Photonic Crystals

Topological photonic crystals (PhCs) that support disorder-resistant modes, protected degeneracies, and robust transport have recently been explored for applications in waveguiding, optical isolation, light trapping, and lasing. However, designing PhCs with prescribed topological properties remains challenging because of the highly nonlinear mapping from the continuous real-space design of PhCs to the discrete output space of band topology. Here, we introduce a machine learning approach to address this problem, employing Kolmogorov--Arnold networks (KANs) to predict and inversely design the band symmetries of two-dimensional PhCs with two-fold rotational (C2) symmetry. We show that a single-hidden-layer KAN, trained on a dataset of C2-symmetric unit cells, achieves high accuracy in classifying the topological classes of the lowest lying bands. We use the symbolic regression capabilities of KANs to extract algebraic formulas that express the topological classes directly in terms of the Fourier components of the dielectric function. These formulas not only retain the full predictive power of the network but also provide novel insights and enable deterministic inverse design. Using this approach, we generate photonic crystals with target topological bands, achieving high accuracy even for high-contrast, experimentally realizable structures beyond the training domain.