Physics Aware Neural Networks: Denoising for Magnetic Navigation

TL;DR

This paper introduces physics-informed neural networks for magnetic anomaly navigation, enforcing Maxwell's laws and E(3)-equivariance to improve magnetic field estimation under noisy conditions.

Contribution

It proposes a novel neural network framework incorporating divergence-free and E(3)-equivariant constraints for better magnetic field modeling in navigation tasks.

Findings

Embedding physical constraints improves prediction accuracy.

The Contiformer model outperforms existing methods.

Synthetic data generation aids in training under data scarcity.

Abstract

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 handle stochastic noise, we propose using two physics-based constraints: divergence-free vector fields and E(3)-equivariance. These ensure the learned magnetic field obeys Maxwell's equation and that outputs transform correctly with sensor position and 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 represented via spherical harmonics with known rotational transformations. Enforcing physical consistency and restricting the admissible function space acts as an implicit regularizer that improves spatiotemporal performance. We present ablation studies evaluating each constraint alone and jointly across CNNs, MLPs, LTCs, and Contiformers. Continuous-time dynamics and long-term memory are critical for modelling magnetic time series; the Contiformer, which provides both, outperforms existing methods. To mitigate data scarcity, we generate synthetic datasets using the World Magnetic Model (WMM) and 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. Acknowledgement: This work was done in collaboration with Dirac Labs.