BrainField-Operator: A Physics-Informed Neural Operator Framework for Bioelectromagnetic Brain Field Mo

Non-invasive brain stimulation techniques such as transcranial direct current stimulation (tDCS) generate weak intracranial electric fields whose spatial distribution depends sensitively on tissue conductivity, anatomical geometry, and electrode placement. Accurately modeling these fields requires solving partial differential equations (PDEs), typically the Poisson equation, which becomes computationally expensive for large parameter sweeps or optimization tasks.

To address this challenge, we introduce BrainField-Operator, a physics-informed neural operator framework for fast approximation of brain electric fields. The framework integrates:
(1) a layered biophysical head model,
(2) a 2D Poisson-based PDE solver,
(3) randomized electrode configurations, and
(4) a Fourier Neural Operator (FNO) surrogate capable of mapping conductivity distributions and electrode masks to potential fields.

The FNO surrogate achieves a mean-squared error below 1.2 × 10⁻⁴ and a relative L₂ error under 10%, demonstrating high fidelity to numerical PDE solutions. This hybrid PDE–ML pipeline enables rapid, resolution-invariant field prediction and forms a foundation for future computational neurotechnology workflows. The framework shows strong potential for accelerating the design and optimization of stimulation protocols in both research and clinical settings.