Solving Poisson’s Equation in Electromagnetics with Limited Data and Arbitrary Domain Deformation Using Physics-enhanced Neural Operator

Modeling complex electromagnetic systems entails the evaluation of partial differential equations (PDEs), a computationally demanding task for conventional high-precision numerical solvers. In recent years, data-driven solvers have emerged as a promising alternative due to their inherent speed advantages during online tests. However, the adoption of data-driven approaches presents several challenges. One such challenge is the acquisition of substantial and high-quality data for effective model training, a process that can be time-consuming and expensive. Another significant difficulty lies in ensuring good generalization performance, as trained models often struggle to perform well when confronted with previously unseen situations.To address these challenges, we propose a physics-enhanced neural operator (PENO) for solving the generalized Poisson’s equation. The PENO solver consists of two key steps. The first step involves employing the Born approximation method as a physical pre-processing technique to rapidly compute an initial approximation solver. In the second step, we construct a neural operator that approximates the mapping between the physics-enhanced input (i.e., the Born approximation solver) and the true solution using a residual structure, where the neural operator model can be flexibly chosen based on specific demands. Moreover, the PENO can be extended to deal with domain-varying problems through incorporating conformal mapping tools. Without the need for re-training, the PENO trained in a region can be transferred to predict solutions of Poisson’s equations defined in different regions.Extensive numerical experiments using benchmark datasets demonstrate the effectiveness of the proposed PENO in reducing the size of the training dataset and enhancing generalizability to unseen scenarios, compared to baseline neural operator solvers such as Fourier neural operator (FNO) and deep operator networks (DeepONet). We anticipate that the proposed PENO learning scheme will find applications in data-driven electromagnetic solvers that require less training data while exhibiting high generalization abilities.