Modeling the inverse MEG problem in neuro-imaging using Physics Informed Neural Networks

Magnetoencephalography (MEG) forward and inverse modeling is fundamental to neuroscientific discovery, yet the inversion of partial differential equations (PDEs) remains one of the most difficult challenges due to its inherent ill-posedness. While traditional numerical methods often struggle with the computational burden and regularization requirements of these problems, neural networks have recently emerged as a highly viable alternative, offering the ability to learn complex, non-linear mappings and provide efficient, real-time inference. This paper presents a framework for the MEG forward and inverse problems, integrating finite element modeling with neural network techniques. The forward problem is solved using FEniCS to model the electric potential governed by the Poisson equation on a realistic anatomical brain mesh, with magnetic fields computed via the Biot-Savart law. For the inverse problem, we introduce a Physics-Informed Neural Network (PINN) approach in order to deal with the ill condition of the problem. Unlike purely data-driven deep learning approaches that treat this problem as a black box learned from massive datasets, the proposed PINN framework directly embeds the governing physics--Maxwells equations and the Biot-Savart law--into the loss function, ensuring that the reconstructed sources satisfy the fundamental electromagnetic laws even in data-scarce regimes. We validate the framework on a high-resolution anatomical mesh and compared against the standard Minimum Norm Estimation (MNE). Results demonstrate that the PINN approach achieves a 30.2% improvement over the MNE baseline.