High precision, full-vector optical mode solving in waveguides via fourth-order derivative physics-informed neural networks.

Optical mode solving plays a critical role in photonic device design, yet conventional numerical methods face inherent challenges, including limited geometric adaptability and computational demands of large-scale matrix eigenvalue problems. Physics-informed neural networks (PINNs) tightly couple neural networks with physical principles, showcasing significant capabilities in photonics for addressing both forward computation and inverse design challenges. This work introduces fourth-order derivative PINNs (4DPINNs) for full-vector waveguide eigenmode solutions. The 4DPINNs simultaneously resolve tangential electric and magnetic field components, enabling direct mode analysis and optical efficiency computations. The network systematically integrates boundary conditions, initialization protocols, and a fourth-order derivative loss function derived from Maxwell's equations. We first validate 4DPINNs by determining electric field distributions for predefined propagation constants, comparing fixed-point initialization strategies with random-point approaches. The solutions achieve maximum absolute errors below -12 dB and minimum absolute errors below -50 dB relative to analytical benchmarks. Through adaptive learning rate optimization, we further demonstrate simultaneous prediction of mode propagation constants and field distributions. The 4DPINNs constrain propagation constant errors to under 10^-4, keeping maximum field distribution absolute errors below -12 dB compared to analytical solutions. Our work demonstrates a highly accurate and broadly applicable waveguide eigensolver, offering substantial value for semiconductor devices and photonic integrated circuits.