BioMedInformatics · Published 2026-07-09 · DOI 10.3390/biomedinformatics6040042
Athanassios S. Fokas, Alireza Afzal Aghaei, Parham Hashemzadeh
The distributed inverse source problem in electroencephalography (EEG) requires the determination of a current-independent, geometry-dependent auxiliary function, which is defined by a Poisson partial differential equation (PDE), where its solution is referred to as the forward problem. In this study, we investigate the feasibility of employing a mesh-free Physics-Informed Neural Network (PINN) for obtaining this auxiliary function. The proposed architecture integrates Kolmogorov–Arnold Networks (KANs) into an extended PINN (XPINN) framework augmented with Multi-scale Fourier feature mappings to capture potential field discontinuities across piecewise-homogeneous tissue interfaces. The PINN loss functional incorporates the governing PDE, Neumann boundary conditions, flux continuity and reference data for specific neuronal source and electrode configurations. Numerical experiments on a three-layer spherical head model demonstrate that the XPIKAN surrogate achieves a relative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mn>2</mn></msub></semantics></math></inline-formula> error below 1% on unseen sensor coordinates. Factorial sensitivity analyses confirm stable model generalization across varying source-sensor configurations without the need for dense volumetric meshes. As a result, XPIKAN provides a meshless, continuous, and differentiable solution that offers faster inference time compared to classical solvers like finite element or boundary element methods and enables exact gradient computation for inverse source localization.
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Fokas, A., Aghaei, A., Hashemzadeh, P. (2026). A Preliminary Analysis of a Physics-Informed Neural Network for the Forward Problem in EEG. BioMedInformatics. https://doi.org/10.3390/biomedinformatics6040042