Physics Aware Neural Networks: Denoising for Magnetic Navigation

Magnetic-anomaly navigation, leveraging small-scale variations in the Earth's magnetic field, is a promising alternative when GPS is unavailable or compromised. Airborne systems face a key challenge in extracting geomagnetic field data: the aircraft itself induces magnetic noise. Although the classical Tolles-Lawson model addresses this, it inadequately handles stochastically corrupted magnetic data required for navigation. To address stochastic noise, we propose a framework based on two physics-based constraints: divergence-free vector field and E(3)-equivariance. These ensure the learned magnetic field obeys Maxwell's equations and that outputs transform correctly with sensor position/orientation. The divergence-free constraint is implemented by training a neural network to output a vector potential $A$, with the magnetic field defined as its curl. For E(3)-equivariance, we use tensor products of geometric tensors representable via spherical harmonics with known rotational transformations. Enforcing physical consistency and restricting the admissible function space acts as an implicit regularizer that improves spatio-temporal performance. We present ablation studies evaluating each constraint alone and jointly across CNNs, MLPs, Liquid Time Constant models, and Contiformers. Continuous-time dynamics and long-term memory are critical for modelling magnetic time series; the Contiformer architecture, which provides both, outperforms state-of-the-art methods. To mitigate data scarcity, we generate synthetic datasets using the World Magnetic Model (WMM) with time-series conditional GANs, producing realistic, temporally consistent magnetic sequences across varied trajectories and environments. Experiments show that embedding these constraints significantly improves predictive accuracy and physical plausibility, outperforming classical and unconstrained deep learning approaches.