Numerical Method for Band Gap Structure and Dirac Point of Photonic Crystals Based on Recurrent Neural Network

In this paper, we propose a recurrent neural network numerical method with the finite element method for partial differential equations to study the band gap structure and Dirac points in two-dimensional photonic crystals. Electromagnetic wave propagation is governed by Maxwell’s equations. We transform the partial differential equations into large-scale generalized eigenvalue problems by spatially discretising them using the finite element method. Compared with traditional numerical computation methods, neural networks can perform high-speed parallel computation. Existing neural network-based eigenvalue solvers are typically restricted to computing extremal eigenvalues of real symmetric matrix pairs. To overcome this limitation, we develop a novel RNN-based numerical scheme tailored for solving the band structure problem in photonic crystals. We validate our method by computing the dispersion relations of photonic crystals with periodic dielectric columns, achieving excellent agreement with the plane-wave expansion method. In addition, we calculate the Dirac points at the center of the Brillouin zone, which is crucial for understanding the unique optical properties of photonic crystals. We determine the precise filling ratios at which these Dirac points appear, thus providing insight into the relationship between geometrical and material parameters and the appearance of Dirac points.