Bayesian Expandable Ambient Magnetic Mapping (BEAMM):
Continual Learning with a Physics-Constrained Bayesian Neural Network

Anonymous Authors

TLDR Neural mapping method for indoor ambient magnetic fields with quantified uncertainty.

Overview

This paper presents a method that overcomes domain‑size and retraining constraints while preserving uncertainty estimates, we propose Bayesian Expandable Ambient Magnetic Mapping (BEAMM), a physics‑constrained Bayesian neural network (BNN) that enforces Maxwell's equations and continually learns a magnetic field map without a fixed set of basis functions.

The following are the list of contributions of this paper in brief:

  1. Expandable Bayesian magnetic mapping: We propose a Bayesian magnetic mapping framework that supports continual domain growth without basis-function reinitialization, addressing a core limitation of fixed-domain GP methods.
  2. Hard physics-structured representation: We introduce a scalar-potential BNN formulation that enforces curl-free structure by construction rather than through soft constraints.
  3. Online robotic deployment: Online robotic deployment demonstrated on a live ground robot equipped with a quantum magnetometer, with the dataset and code released publicly for reproducibility and future benchmarking.We develop a practical Bayesian inference and continual learning strategy and validate it on a live ground robot equipped with a hybrid quantum magnetometer, with code and datasets released publicly.
An overview video is given below:

Results

We explore the impact of a Bayesian neural network on the magnetic mapping problem when compared against traditional Gaussian process based methods. Several ablation studies were conducted initially to understand the modeling effects of the chosen parameters. An overview of parameters are given below:

Parameter Value
Hidden Neurons 128
Number of NUTS Samples 50
Initial NUTS Step Size ($\tau$) $3 \times 10^{-6}$
NUTS exploration steps per sample 30
Number of Hidden Layers 2
Activation Function Softplus

Ablations

First the hybrid prior for cold starting and reduced sample requirement are ablated and tested:



The NUTS samples are then ablated to show the faster convergence with reduced samples


Dataset Head-to-Head Comparison

The framework was compared head to head on two other SoTA architectures. (i) A Hilbert space gaussian process (G-Hilbert) and (ii) a deterministic training conditional Gaussian process (G-DTC). The three different datasets used here that can be found at https://github.com/CEA-LIST/large-scale-magnetic-mapping

Dataset Method MSE ($\mu\mathrm{T}^2$) MSLL Prediction Time ($\mathrm{ms}$)
Simu1D G-DTC $7.7\times 10^{-5}$ $-11.8$ $2.2$
G-Hilbert $7.8\times 10^{-5}$ $-12.3$ $28$
BEAMM $1.84\times 10^{-4}$ $-11.5$ $8.2$
Simu2D G-DTC $1.9 \times 10^{-4}$ $-11.6$ $8.7$
G-Hilbert $1.9 \times 10^{-4}$ $-11.7$ $28$
BEAMM $3.8 \times 10^{-4}$ $-10.2$ $7.3$
Simu3D G-DTC $4.9\times 10^{-4}$ $-11.1$ $43$
G-Hilbert $5.1\times 10^{-4}$ $-10.9$ $28$
BEAMM $9.5\times 10^{-4}$ $-9.7$ $11.2$

Hardware Tests

Finally we also ran the framework on hardware to see if the system can be implemented and deployable.

Field magnitude map
Final map with field magnitude in $\mu\mathrm{T}$.
Coverage vs time
Bounding box coverage vs. time.
Snapshot c Snapshot d Snapshot e Snapshot f
(c) --- (f) Sequential snapshots at $T=0,~110,~245,~\text{and}~ 620~\mathrm{s}$.

A demonstration of the framework on physical hardware is shown below. The video is sped up 10x as the robot is moving at $0.2 ~\mathrm{m/s}$ as the mapping is on average slow moving.

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