Modeling the inverse MEG problem inneuro-imaging using Physics Informed NeuralNetworks

Magnetoencephalography (MEG) forward and inverse modeling is fundamen-tal 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 computa-tional burden and regularization requirements of these problems, neural networkshave recently emerged as a highly viable alternative, offering the ability to learncomplex, non-linear mappings and provide efficient, real-time inference. Thispaper presents a framework for the MEG forward and inverse problems, integrat-ing finite element modeling with neural network techniques. The forward problemis solved using FEniCS to model the electric potential governed by the Poissonequation on a realistic anatomical brain mesh, with magnetic fields computed viathe Biot-Savart law. For the inverse problem, we introduce a Physics-InformedNeural Network (PINN) approach in order to deal with the ill condition of theproblem. Unlike purely data-driven deep learning approaches that treat this prob-lem as a black box learned from massive datasets, the proposed PINN frameworkdirectly embeds the governing physics—Maxwell’s equations and the Biot-Savartlaw—into the loss function, ensuring that the reconstructed sources satisfy thefundamental electromagnetic laws even in data-scarce regimes. We validate theframework on a high-resolution anatomical mesh and compared against the stan-dard Minimum Norm Estimation (MNE). Results demonstrate that the PINNapproach achieves a 30.2% improvement over the MNE baseline.