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Physics Aware Neural Networks: Denoising for Magnetic Navigation

Aritra Das, Yashas Shende, Muskaan Chugh, Reva Laxmi Chauhan, Arghya Pathak, Debayan Gupta

TL;DR

This work tackles denoising of magnetometer data for magnetic anomaly navigation in GNSS-denied contexts, where aircraft-induced noise contaminates the geomagnetic signal. It introduces physics-aware neural networks that enforce a divergence-free magnetic field via $\mathbf{B}=\nabla\times\mathbf{A}$ and $E(3)$-equivariance using spherical harmonics, combined with continuous-time neural ODEs and a long-term memory latent state. To address data scarcity, synthetic sequences are generated with World Magnetic Model-based conditioning using a conditional GAN. Experiments across multiple backbones show that embedding these physical priors yields substantial performance gains over classical and unconstrained deep-learning approaches, highlighting the value of physics-informed design for robust spatio-temporal magnetic denoising in navigation systems.

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 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.

Physics Aware Neural Networks: Denoising for Magnetic Navigation

TL;DR

This work tackles denoising of magnetometer data for magnetic anomaly navigation in GNSS-denied contexts, where aircraft-induced noise contaminates the geomagnetic signal. It introduces physics-aware neural networks that enforce a divergence-free magnetic field via and -equivariance using spherical harmonics, combined with continuous-time neural ODEs and a long-term memory latent state. To address data scarcity, synthetic sequences are generated with World Magnetic Model-based conditioning using a conditional GAN. Experiments across multiple backbones show that embedding these physical priors yields substantial performance gains over classical and unconstrained deep-learning approaches, highlighting the value of physics-informed design for robust spatio-temporal magnetic denoising in navigation systems.

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 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 , 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.
Paper Structure (18 sections, 19 equations, 4 figures, 2 tables)

This paper contains 18 sections, 19 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work and Preliminaries
  3. Filtering Based Approaches
  4. Tolles-Lawson Model
  5. Deep-Learning based Approaches
  6. Our Approach
  7. Divergence-free constraint
  8. E(3)-equivariance
  9. Continuous-time handling
  10. Long-Term memory
  11. Synthetic Data Generation using Conditional GaN
  12. Experiments
  13. Data Pre-Processing
  14. Training/Testing
  15. Baselines and Metrics
  16. ...and 3 more sections

Figures (4)

  • Figure 1: An illustrative diagram of our problem setting.
  • Figure 2: Maxwell's 2nd Law of Electromagnetism. Magnetic field lines have no beginning or end, but exist as loops or extend to infinity: magnetic monopoles cannot exist.
  • Figure 3: E(3)-equivariance demonstration: applying geometric transformations before or after the neural network $\varphi$ yields equivalent results, ensuring consistent magnetic field predictions regardless of coordinate system orientation.
  • Figure 4: Denoising Pipeline with Divergence-Free and E(3)-Equivariant Constraints