Neural network enhanced finite difference method for magnetostatic simulations: A localized, context-aware approach for bridging finite difference and finite element methods accuracy

This work introduces a hybrid computational framework for the efficient and accurate solution of magnetostatic boundary value problems by combining the Finite Difference Method (FDM), the Finite Element Method (FEM), and a Deep Neural Network (DNN). The proposed approach leverages a coarse-grid FDM solution to provide a fast initial approximation, which is subsequently enhanced through a learned correction that maps FDM outputs to FEM-level accuracy. The core contribution lies in a localized, cell-wise training strategy in which the DNN is trained using FDM-derived features and high-fidelity FEM solutions as targets, enabling the model to capture both local field behavior and global structure through the incorporation of Fourier-based contextual information. Unlike classical numerical methods, which require a full recomputation for each new configuration, the proposed framework learns a parametric mapping with respect to the current density, allowing efficient and straightforward generalization across different source terms without re-solving the governing equations. This characteristic could make the method also a practical alternative to physics-informed neural networks (PINNs), which typically require complex training procedures to achieve similar parametric capabilities.