Geometry–Condition Factorized Neural Operators for Electromagnetic Modeling

Electromagnetic forward modeling is fundamental to applications ranging from mineral exploration to cerebral electrical impedance tomography (EIT), yet repeated high-fidelity simulations remain computationally expensive. Starting from the Maxwell-governed forward operator, we analyze its local Fr\'echet sensitivity with respect to the three-dimensional electrical-property field. The resulting sensitivity exhibits a pronounced geometry–condition organization: the corresponding unfolding shows substantially stronger spectral concentration than alternative variable partitions, indicating that observation geometry and excitation condition admit distinct yet coupled representations. Guided by this structure, we propose the Geometry–Condition Factorized Neural Operator (GCFNO). A shared volumetric encoder extracts the latent electromagnetic state, a geometry factor models receiver interactions through coordinate-conditioned message passing, while a condition factor performs frequency-guided aggregation of depth-resolved features. A nonlinear readout then couples the two factors to predict sensor responses. On 3-D magnetotelluric modeling, GCFNO achieves the lowest three-seed mean total RMSE, while substantially improving held-out-frequency and receiver-geometry generalization. On head EIT, GCFNO achieves the lowest three-seed mean patient-macro error among evaluated surrogates. More broadly, these results suggest a path toward problem-specific neural operators whose architectures are derived from the governing physics, enabling more accurate and generalizable surrogates for complex physical systems. The methods and related analysis code of this manuscript are available at https://anonymous.4open.science/r/iclr2027-code-artifact-788E.