Physics-informed neural operators for rapid electric field prediction in transcranial magnetic stimulation of healthy and tumor subjects

Accurate and rapid prediction of transcranial magnetic stimulation (TMS)-induced electric fields is essential for personalized clinical applications, from individualized therapy in major depressive disorder to presurgical functional mapping in neuro-oncology. Yet conventional finite element methods are often too computationally expensive for interactive use. Existing deep learning surrogate models partly address this limitation but are largely constrained to fixed voxel-grid representations that cannot naturally accommodate the irregular geometry of subject-specific head meshes. We present the first physics-informed neural operator (PINO) for subject-specific prediction of vector TMS electric fields, built on a mesh-native dual-encoder architecture that jointly encodes head anatomy and coil configuration. By incorporating a curl-based regularization term aligned with the curl-free nature of the TMS secondary field, the PINO model promotes physical consistency as a soft regularizer alongside data-driven supervision. Trained on 75 subjects from the IXI dataset, the framework was evaluated across four independent held-out test datasets, including healthy subjects from two additional MRI cohorts and, for the first time in learning-based TMS electric-field prediction, tumor patients. The resolution-agnostic formulation generalizes across subjects, coil types, and spatial resolutions without retraining, predicting vector electric fields on approximately 300,000 gray matter volume nodes in under 100 ms on a consumer-grade GPU. Across all test datasets within the focality-50% region, the PINO model achieved median center-of-gravity errors of 0.39–0.43 mm, RMSE of 0.12–0.16 V/m, and mean angular deviations of 6.7°–7.3°, with performance within or below the error ranges reported by recent deep-learning approaches. At the selected 5% physics-loss weighting, curl loss decreased by 9.1% relative to the data-only model, while RMSE and MAD increased by only 1.6% and 1.3%, respectively. These results suggest that physics-informed neural operators provide a scalable technical foundation for rapid gray-matter electric-field approximation, supporting future studies on real-time electric-field-guided TMS workflows