Gaussian Process Regression for Magnetic Field Reconstruction

https://slides.com/qudsi/magnetore

Some relevant scales:

d_i \mathrm{(1AU)} \sim 100 ~\mathrm{km} \\ V_{sw} \sim 500 ~\mathrm{km/sec} \\ X_{sim} \mathrm{(box size)} \sim 40 d_i\\ \hspace{3.1cm}\sim 4\times 10^3 \mathrm{km}\\ d_{spc} \sim [1, 11]~d_i\\ \hspace{1.2cm}\sim \mathrm{ [10^2, 10^3]~ km} \\ f[min,max] \sim V_{sw}/(2\times d_{spc, max, min}) \\ \hspace{0.8cm}\sim [0.25, 2.5]~\mathrm{Hz}\\ \nu (\mathrm{sampling~rate}) \sim 40 \mathrm{Hz}\\ \mathrm{(just~because~of~box~size~and~stuff)}\\ \mathrm{time~to~move~across~the~simulation~box} \sim 10 ~\mathrm{sec}

Hub

Spacecrafts

$$d_i$$

22

Gaussian Process Regression Interpolation

Gaussian Process Regression:

It is a probabilistic data imputation method

It is a probability distribution over possible functions that fit the given finite dataset

dataset with finite number of observation is modelled as if it were a multivariate normal distribution

$$m(\mathbf{x}) = \mathbb{E}[f(\mathbf{x})]$$

Mean function

$$k(\mathbf{x}, \mathbf{x'}) = \mathbb{E}[(f(\mathbf{x}) - m(\mathbf{x}))(f(\mathbf{x'}) - m(\mathbf{x'}))]$$

Covariance function

$$f(\mathbf{x}) \sim \mathcal{GP}\left(m(\mathbf{x}), k(\mathbf{x}, \mathbf{x'})\right)$$

Gaussian Processes

Constant

Linear

RBF

Matern

Kernels

Results

One component at a time

Maruca 2021, Frontiers in A & SS

All components at a time

Qudsi et. al., in prep

Single component input

Vector input

simulation data

simulation data

8 spc

24 spc

8 spc

24 spc

8 spc

16 spc

24 spc

34 spc

24 spc*

22 spc*

Moving Forward

What's next?

Explore the impact of relative spacecraft separation and arrangement

Incorporate the zero divergence condition on magnetic field

Explore alternative algorithms/methods

Deep Gaussian Processes

Neural Network as GP

Find ways to compare the reconstructed images for different styles

Thank You!