A Physics-Informed Neural Networks for Solving 2D Frequency-Domain Electromagnetic Waves in the Differential Form of Maxwell’s Equations

This article investigates the application of physics-informed neural networks (PINNs) for solving two-dimensional (2D) frequency-domain Maxwell’s equations in complex geometries, including rectangular, circular, annular, and eccentric circular domains. The problem is reformulated into a system of Helmholtz equations governing the electric and magnetic fields in the frequency domain. The proposed PINN framework incorporates the governing equations, boundary conditions, and the complex-valued nature of the problem directly into the loss function by decomposing the field components into real and imaginary parts. The performance of the method is evaluated against analytical solutions and compared with the finite difference method (FDM). Numerical results demonstrate that the PINN consistently achieves lower relative L2 errors and produces smoother and more accurate solutions, particularly in annular and eccentric domains with internal and irregular boundaries, where the FDM exhibits noticeable discrepancies. Furthermore, comparisons with advanced PINN variants indicate that R-PINNs improve local accuracy through adaptive sampling, while attention-based PINNs enhance performance via dynamic feature weighting. The proposed method surpasses these approaches, achieving lower MSE and significantly improved accuracy, with errors reduced to the order of 10−7–10−8. Overall, the results highlight the robustness, accuracy, and geometric flexibility of PINNs for solving frequency-domain Maxwell’s equations in complex domains.