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 The authors acknowledge Claude (Anthropic) for assisting with language editing and improving clarity. This work is financed by research subsidies granted by the government of Upper Austria (research projects MIMAS.ai, mimas.ng) and by the FFG (research project nARvibrain, grant no. 894756). www.ffg.at. RISC Software GmbH is a member of UAR (Upper Austrian Research) Innovation Network.

Research Unit Medical Informatics, RISC Software GmbH, Hagenberg, 4232, Austria Philipp Moser, Fabian Greifeneder, Dominik Freinberger & Stefan Thumfart Department of Radiology, Medical University of Graz, Graz, 8036, Austria Department of Radiology, Clinical Division of Neuroradiology, Vascular and Interventional Radiology, Medical University of Graz, Graz, 8036, Austria Department of Neurosurgery, Medical University of Graz, Graz, 8036, Austria The authors declare no competing interests. The use of data from both healthy subjects (MUG-TMS) and tumor patients (MUG-Tumor) collected at the Medical University of Graz was approved by the local ethics committee within the framework of the nARvibrain research project (approval number: ECS 115/2024). Informed consent was obtained from all subjects.

The publicly available IXI Guy’s Hospital and Kirby21 datasets were used in accordance with their respective data usage terms. All procedures were conducted in accordance with the ethical principles of the Declaration of Helsinki. Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.