PINNs for solving unsteady Maxwell’s equations: convergence issues and comparative assessment with compact schemes

Physics-Informed Neural Networks (PINNs) have recently gained prominence as a mesh-free, physics-integrated framework for solving partial differential equations. In this work, we evaluate the capabilities of PINNs in solving unsteady Maxwell’s equations, benchmarking their performance against two established numerical schemes: the finite-difference time-domain method and a compact Padé scheme. The investigation spans three canonical test cases, involving one-dimensional free-space wave propagation and two-dimensional Gaussian pulse evolution in both periodic and dielectric media. The convergence enhancement strategies including random Fourier feature embeddings, spatio-temporal periodicity enforcement, and temporal causality constraints are assessed systematically through ablation study. The results suggest that architectural design choices must be closely aligned with the governing physics to ensure stable and accurate convergence. Using Neural Tangent Kernel analysis, the intrinsic “uneven learning” behavior of PINNs is uncovered. It was observed that PINNs can fail to prioritize regions with high error, instead converging more rapidly where the loss is already small, contrary to effective optimization principles. Overall, this study demonstrates that PINNs, with appropriate architecture, can match or surpass numerical solvers in accuracy and flexibility. However, challenges remain in addressing spatial inhomogeneity of convergence rate (uneven learning), adapting training to localized high-gradient features, and computational cost.