Electro-Diffusion Physics-Informed Neural Network (ED-PINN) for 3D Brain Electric Field Reconstruction from EEG Signals

Reconstructing the continuous, three-dimensional (3D) distribution of intracranial electric potential from sparse, non-invasive scalp electroencephalography (EEG) is a central inverse problem in computational neuroimaging. This work introduces the Electro- Diffusion Physics-Informed Neural Network (ED-PINN), a coordinate-based neural representation that enforces the governing quasi-static Maxwellian electro-diffusion equation, ∇ · (σ(x)∇ϕ(x)) = −I(x), as a soft constraint during training. By parameterizing the potential field ϕ(x) as a continuous function, ED-PINN integrates sparse electrode measurements, Dirichlet/Neumann boundary conditions, and collocation-based PDE residuals into a single, unified objective function. This mesh-free approach enables the reconstruction of physically consistent, differentiable volumetric fields without the need for explicit domain meshing required by traditional methods like FEM or BEM. We demonstrate the efficacy of this approach on a canonical three-layer spherical head model with realistic tissue conductivities and synthetic Gaussian sources. We present a quantitative and qualitative evaluation, analyze primary error sources, and outline a clear roadmap for extensions to anatomically realistic geometries and anisotropic conductivity tensors derived from diffusion MRI. The experiments show that ED-PINN produces smooth, differentiable potential fields and localizes sources to sub-centimeter accuracy under the studied conditions. The paper includes detailed implementation notes, training recipes suitable for Colab/CPU environments, and a curated bibliography to ensure reproducibility.