Title: Physics Aware Neural Networks : Denoising for Magnetic Navigation
Primary area: learning on time series and dynamical systems
Keywords: irregular time-series, physics-aware neural networks, magnetic navigation

Abstract: Physics Aware Neural Networks : Denoising for Magnetic Navigation

Magnetic-anomaly navigation, which leverages small-scale local variations in the Earth's magnetic field, has emerged as a promising alternative for environments where GPS signals are unavailable or compromised. Airborne systems face a fundamental challenge in extracting the necessary geomagnetic field data: the aircraft itself induces magnetic noise, which needs to be removed. While the classical Tolles-Lawson model addresses this, it inadequately handles stochastically corrupted magnetic data needed for operational navigation.
To handle stochastic noise, we propose a novel approach using two physics-based constraints: divergence-free vector fields and E(3)-equivariance. Our constraints guarantee that the generated magnetic field obeys Maxwell's equations, and ensure that outputs change appropriately with sensor position/orientation. The divergence-free constraint is implemented by defining a neural network outputting vector potential A, with the magnetic field constructed as its curl. For E(3)-equivariance, we use tensor products of geometric tensors representable using spherical harmonics with well-known rotational transformations. By enforcing physical consistency and constraining the space of admissible field functions, our formulation acts as an implicit regularizer, improving spatiotemporal performance. 
We conduct ablation studies evaluating these constraints' individual and combined effects across CNNs, MLPs, Liquid Time Constant Models, and Contiformers. We note that continuous-time dynamics and long-term memory are critical for modelling magnetic time-series data; the Contiformer architecture, which inherently possesses both, surpasses state-of-the-art methods in our experiments. To handle data scarcity, we develop synthetic datasets by utilising the World Magnetic Model (WMM) in conjunction with time-series conditional GANs, generating realistic and temporally consistent magnetic field sequences spanning various trajectory patterns and environmental scenarios. Our experiments demonstrate that embedding these constraints significantly improves predictive accuracy and physical plausibility outperforming the state-of-the-art across both classical and unconstrained deep learning approaches.

TL;DR: Closed-form solution for Contiformer Using Damped Harmonic Oscillators