A Fourier feature physics-informed neural network framework for accurate mesh-free solutions of two-dimensional time-domain Maxwell equations

Physics-informed neural networks (PINNs) provide a mesh-free framework for solving partial differential equations but are limited by spectral bias when approximating high-frequency solutions. This work investigates the Fourier feature PINNs (FF-PINNs) for solving the two-dimensional time-domain Maxwell equations in transverse electric mode. The system is reformulated as hyperbolic type equations governing the electric and magnetic field components, where the electromagnetic fields (Ex, Ey, Hz) are approximated using fully connected neural networks trained through a composite physics-informed loss that enforces governing equations and initial and boundary conditions. To improve the representation of oscillatory wave behavior, Fourier feature embeddings are incorporated into the FF-PINN model, and a hybrid Adam–L-BFGS optimization strategy is employed to enhance convergence. Numerical experiments are performed on rectangular, circular, and L-shaped domains to evaluate performance under both smooth and irregular geometries. The results show that standard PINNs achieve reasonable solutions but exhibit reduced accuracy and slower convergence in regions with high-frequency oscillations and geometric singularities. In contrast, FF-PINNs significantly improve accuracy, convergence, and stability, yielding lower relative L2 errors and smoother field distributions, particularly near curved boundaries and re-entrant corners. Overall, the FF-PINN framework offers a robust and accurate mesh-free approach for electromagnetic wave simulations and shows strong potential for extension to higher-dimensional and frequency-domain Maxwell problems.