Physics-Informed Neural Networks and Graph Neural Networks for the Numerical Modeling of the Vector Helmholtz Equation

This workinvestigates neural network representations of vector fields governed by the vector Helmholtz equation, with particular attention on domains containing discontinuous material properties. The de Rham complex and its discrete Whitney–Nédélec counterpart are considered to clarify the relationship between the regularity of neural representations and the natural function spaces of electromagnetic vector fields. Three FEM-supervised neural approaches are studied: a vector PINN for a homogeneous domain, an XPINN for a heterogeneous domain with a material interface, and an edge-based graph neural network for predicting Nédélec edge circulations. In the heterogeneous formulation, the XPINN interface loss is constructed from the physically appropriate transmission conditions: continuity of the tangential electric field, continuity of the curl-related flux, and continuity of the normal electric displacement (D=εE),while allowing the normal component of (E) itself to be discontinuous when the material parameters change. The numerical results show that smooth neural representations can approximate vector fields in both homogeneous and heterogeneous media when the relevant (H(curl)) boundary and interface conditions are incorporated explicitly. The edge-based GNN directly predicts the lowest-order Nédélec circulation degrees of freedom on mesh edges and reproduces the corresponding FEM solution with a relative (L2) error of (2.86%). The revised XPINN achieves a full-domain relative (L2) error of (3.56%) with respect to the FEM reference. These results demonstrate the feasibility of FEM-supervised neural surrogates for continuous or edge-based representation of vector Helmholtz solutions. Once trained for a fixed problem instance, the resulting neural models provide inexpensive field evaluation without repeatedly solving the associated sparse finite element system.