Advances in PDE-Constrained Optimization for Electromagnetic Field Control: A Review of Theory and Applications

Partial Differential Equation (PDE)-constrained optimization has emerged as a powerful framework for electromagnetic field control, enabling systematic design of devices and materials that meet stringent performance, efficiency, and reliability requirements. This review traces the development of PDE-based formulations grounded in Maxwell’s equations, highlighting discretization strategies, adjoint-state methods, and large-scale solvers that make high-dimensional optimization problems computationally tractable. Key applications are examined across antenna design, electromagnetic compatibility and shielding, nanophotonics, metamaterials, biomedical imaging, and emerging quantum technologies. These studies illustrate how PDE-constrained optimization bridges physics-based modeling with engineering innovation, achieving designs that were previously inaccessible through heuristic or trial-and-error approaches. Despite rapid progress, challenges persist in scalability, nonconvex optimization landscapes, uncertainty quantification, and multiphysics integration. Recent advances in reduced-order modeling, surrogate-assisted optimization, and robust design strategies offer promising avenues to overcome these limitations. Furthermore, new computational paradigms, particularly high-performance computing and data-driven surrogates, are reshaping possibilities for solving complex, nonlinear electromagnetic problems at scale. Overall, PDE-constrained optimization has matured into a versatile and rigorous approach with significant implications for next-generation communication, energy, biomedical, and quantum technologies. Continued methodological and computational advancements will be critical for realizing its full potential in real-world electromagnetic design.