A Preliminary Analysis of a Physics-Informed Neural Network for the Forward Problem in EEG

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 $L_2$ 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.