Robust Physics-Informed Penalty-Free Machine Learning Method for Solving 3-D Maxwell’s Equations in the Frequency Domain

In this work, we tackle the research problem of robust and efficient neural network (NN)-based solutions of 3-D Maxwell’s equations in the frequency domain, subject to boundary conditions. We formulate an unconstrained optimization problem that, by construction, satisfies arbitrary boundary conditions—whether Neumann, Dirichlet, or Robin—regardless of the geometry and material properties. This removes the need to introduce penalty terms. The optimization problem formulated is quadratic, and its strong convexity ensures that convergence to the optimum point is guaranteed, even with gradient descent-based algorithms. In addition, we design an NN architecture with batch normalization (BN), residual net shortcuts, and multitask learning to improve training performance. For ill-conditioned problems, we reformulate them into well-conditioned ones to ensure convergence. To further reduce the computational cost, we accelerate the dense matrix-vector multiplication (MVM) involved in the loss function evaluation using advanced${\mathcal {H}}^{2}$-techniques. The proposed ML method is guided solely by physical laws without the need for data, and it has been successfully applied to solve surface and volume integral equations (IEs), as well as partial differential equations (PDEs), for solving complex 3-D electromagnetic (EM) problems in the frequency domain. Numerical experiments and comparisons with existing solvers have demonstrated its accuracy, robustness, and generality.