Structured Learning for Electromagnetic Field Modeling and Real-Time Inversion

TL;DR

The paper tackles real-time, accurate magnetic field modeling for electromagnetic navigation systems (eMNS) by preserving an affine, current-linear forward map while enabling fast minimum-norm inversion. It introduces two neural architectures, ActuationNet and PotentialNet, that predict the field actuation matrix and curl-free potentials respectively, and compares them against the physics-based MPEM and unstructured baselines. Key findings show data-driven, structure-preserving models achieve predictive fidelity on par with MPEM, maintain strong data efficiency down to roughly 20% of training data, and deliver sub-millisecond evaluation times, with Maxwell-consistency and gradient information preserved. A geometric error floor (~0.5 mT) dominates residuals, rather than model capacity, and appropriate near-field sampling mitigates workspace ill-conditioning with higher-order expansions helping further; the authors also release code and dense datasets to support further research and deployment in both medical and broader electromagnetic contexts.

Abstract

Precise magnetic field modeling is fundamental to the closed-loop control of electromagnetic navigation systems (eMNS) and the analytical Multipole Expansion Model (MPEM) is the current standard. However, the MPEM relies on strict physical assumptions regarding source symmetry and isolation, and requires optimization-based calibration that is highly sensitive to initialization. These constraints limit its applicability to systems with complex or irregular coil geometries. This work introduces an alternative modeling paradigm based on multi-layer perceptrons that learns nonlinear magnetic mappings while strictly preserving the linear dependence on currents. As a result, the field models enable fast, closed-form minimum-norm inversion with evaluation times of approximately 1 ms, which is critical for high-bandwidth magnetic control. For model training and evaluation we use large-scale, high-density datasets collected from the research-grade OctoMag and clinical-grade Navion systems. Our results demonstrate that data-driven models achieve predictive fidelity equivalent to the MPEM while maintaining comparable data efficiency. Furthermore, we demonstrate that straightforward design choices effectively eliminate spurious workspace ill-conditioning frequently reported in MPEM-based calibration. To facilitate future research, we release the complete codebase and datasets open source.

Figures (7)

• Figure 1: Visualization of the positions where magnetic field data has been acquired. (a) Measurement points for the Octomag system. (b) Measurement points for the clinically ready Navion system. To the best of our knowledge, these represent the densest datasets recorded for such systems.
• Figure 2: Validation of magnetic field linearity. (a) Boxplots of fit quality ($R^2$) and field offset magnitudes ($\|\bm{b}_0\|$) for the OctoMag and Navion. (b--c): Representative magnetic field measurements versus commanded current for the OctoMag (b) and Navion (c), demonstrating the affine relationship between current and field generation in both systems.
• Figure 3: Field prediction performance and data efficiency analysis. (a) Test RMSE versus model complexity rank (I, II, III). All model classes converge to an error floor of approximately 0.5 mT, demonstrating that performance is largely insensitive to increased model capacity. (b) Generalization gap (difference between test RMSE and train RMSE) versus complexity rank. The gap remains near zero for most methods, indicating minimal overfitting with the full dataset. (c) Test RMSE as a function of dataset percentage. Predictive accuracy remains stable for most models down to 20% of the training data. (d) Generalization gap as a function of dataset percentage. At low data regimes (1–5%), unstructured baselines (DirectNet, DirectGBT) exhibit significant overfitting, while structured models maintain a smaller generalization gap.
• Figure 4: Geometric sensitivity analysis of the residual error floor. The heatmap displays the expected field prediction error ($\mathrm{RMSE}_{\mathrm{floor}}$) as a function of isotropic sensor position uncertainty ($\sigma_p$) and orientation uncertainty ($\sigma_\theta$). The red iso-contour at 0.5 mT corresponds to the convergence limit observed in \ref{['fig:FieldPred']}, demonstrating that the residual errors in model predictive capability are effectively explained by realistic geometric tolerances (sub-millimeter position and degree-scale orientation offsets) rather than model capacity limitations.
• Figure 5: Evaluation of physical consistency. Boxplots showing the distribution of divergence (top) and curl magnitude (bottom) computed on the test set. The structured learning-based models (ActuationNet, PotentialNet) demonstrate significantly better adherence to Maxwell's equations compared to the unstructured baselines (DirectNet, DirectGBT). Notably, the PotentialNet exhibits zero curl error, confirming that it satisfies the curl-free condition by construction. The MPEM baseline is excluded from this plot as it inherently satisfies both conditions by construction.