Symbolic Learning of Topological Bands in Photonic Crystals

TL;DR

The paper tackles inverse design of two-dimensional photonic crystals to realize prescribed topological bands under $C_2$ symmetry. It trains a lightweight Kolmogorov–Arnol’d network on Fourier components of the dielectric function to predict the lowest-band topology with >99% accuracy and then extracts eight analytic formulas via symbolic regression that map three salient Fourier components to topological classes, enabling deterministic inverse design. The authors demonstrate high-fidelity design for smooth and high-contrast two-tone PhCs, with accuracy >92% even at dielectric contrasts up to 12, and provide concrete examples of Dirac-point and edge-state configurations. This work delivers interpretable design rules, connects machine-learning predictions to perturbation theory, and offers scalable pathways to extend topological photonics design to more complex symmetries and higher bands.

Abstract

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.

Figures (10)

• Figure 1: (a) Dielectric profile of a typical photonic crystal in our dataset, depicted in the basis of lattice vectors, $\mathbf{R}_1, \mathbf{R}_2$. Our dataset is augmented four-fold by considering shifts of the unit cell center to four $C_2$ symmetric Wyckoff positions: 1a, 1b, 1c and 1d (see subfigure (c)), which change the symmetry eigenvalues according to \ref{['equation; data augmentation']}. In the left column of (a), we show the same photonic crystal at these four choices of centering and in the right column we show the corresponding symmetry eigenvalues at the four high symmetry points in the Brillouin zone, with blue and red circles denoting $+1$ and $-1$$C_2$ eigenvalues, respectively. (b) TM-polarized band dispersion of the PhC in (a). The first band is highlighted in blue. We indicate the $C_2$ symmetry eigenvalues of the first band, corresponding to centering at the 1a Wyckoff position. (c) Position of the Wyckoff positions (yellow squares) in the unit cell overlaid on the PhC centered at 1a. (d) High symmetry points in the Brillouin zone (green circles) shown in the basis of primitive reciprocal lattice vectors, $\mathbf{G}_1, \mathbf{G}_2$.
• Figure 2: Kolmogorov--Arnold network trained on $C_2$ symmetric PhCs. The pruned network only depends on three input parameters, with only four nodes in the hidden layer. Three trained activation functions are highlighted in red, green, and blue and shown in more detail in the insets. Points lying on top of the activation functions denote samples from the dataset. We show the decomposition of the dielectric function of the PhC from \ref{['figure: 1']} into the lowest three Fourier components, which comprise the three input parameters. The symmetry classes are labeled by the same convention as in \ref{['figure: 1']}(a). Classes with an odd (even) number of red circles correspond to topological (trivial) bands. We show the trivial (topological) classes in orange (turquoise).
• Figure 3: (a) Inverse design workflow. We first choose a target category and pick a random PhC to serve as a starting point. We next do gradient descent on the PhC's Fourier components to yield a smooth lattice within the target category. We then used \ref{['equation: two-tone-mapping']} to map our smooth PhCs to two-tone PhCs. (b) Accuracy of our inverse design scheme. Blue (yellow) bars correspond to inverse design from the KAN (formulas), respectively. We achieve greater than 99 percent accuracy in creating smooth lattices and low-contrast two-tone lattices. Increasing the dielectric contrast up to 12 lowers the accuracy only moderately, to 92 percent. (c) Band structures at each stage of the inverse design workflow. $C_2$ Symmetry eigenvalues for the lowest band remain unchanged throughout the workflow and are shown overlaid on the rightmost band structure.
• Figure 4: Inverse designed Dirac points and edge states. (a) Band structure corresponding to one of the inverse designed PhCs with nontrivial topology below the first gap. Dashed red line denotes the frequency of the Dirac degeneracy, which occurs in the interior of the Brillouin zone. (b) Band structure corresponding to a one-dimensional tiling of an atomic limit. The edge state, denoted by red, arises from the difference in polarization between the bulk and the trivial cladding of the tiling. Here, the cladding is a shifted version of the same unit cell, obtained using \ref{['equation; data augmentation']}.
• Figure S5: KANs for prediction of band topology beyond the lowest TM band: (a) KAN for prediction of the band symmetries of the second TM band. (b) KAN for prediction of the band symmetries of the first transverse-electric polarized band. In each case, Fourier components given in the $\mathbf{G}_1, \mathbf{G}_2$ basis are mapped to the possible symmetry classes, of which there are sixteen for the second TM band and eight for the first TE band. We delineate trivial vs topological classes by labeling the former with orange and the latter with turquoise dots.