FD-hPINNs-Based Inverse Design Method for Metallic Metastructure

Physics-informed neural networks (PINNs) have attracted widespread attention for the integration of physical constraints and data-driven learning. However, mainstream PINNs frameworks rely on automatic differentiation (AD) to compute partial derivatives, leading to unstable convergence, rapidly increasing computational cost with network scale, difficulties in balancing loss terms, and limited structural expressiveness, constraining applicability in complex electromagnetic (EM) design tasks. A finite-difference hard-constraint physics-informed neural network (FD-hPINNs) is proposed for the inverse design of EM devices. The approach replaces AD with finite-difference (FD) for derivative computation, enhancing the physical correlation among spatial sampling points and effectively reducing the high computational cost that grows with network scale, while maintaining compatibility with various network architectures. In addition, boundary conditions and structure parameters are directly embedded into the partial differential equations (PDEs) computation as hard constraints, simplifying the loss-function composition and improving the adaptability to complex structures. The proposed method is validated with a metallic metastructure in 2-D free space, and the results are compared with those obtained from AD-PINNs, demonstrating the enhanced capability of the FD-hPINNs in complex EM inverse design problems. Furthermore, the design is reimplemented and experimentally verified on a dielectric substrate, where the measurements match well with the simulations, confirming the effectiveness of the FD-hPINNs in practical EM device inverse design.