Reconstructing the Magnetic Field in an Arbitrary Domain via Data-driven Bayesian Methods and Numerical Simulations

TL;DR

This work tackles inverse problems where boundary-condition parameters are unknown and data are sparse. It develops a data-driven Bayesian framework that combines priors, a likelihood based on the forward model, and MAP optimization to infer boundary-condition statistics and reconstruct the magnetic field in an arbitrary domain. The forward problem is solved with finite element methods, including unconstrained and divergence-free constrained formulations via a Lagrange multiplier, and the reconstruction is demonstrated in a conical domain with both single- and multi-prior data, highlighting robustness to data sparsity and providing uncertainty quantification. The approach is scalable, versatile across PDFs, and poised for extension to galactic magnetic-field reconstruction (GMF) and other applications, with planned comparisons to MCMC and broader Bayesian machine-learning methods.

Abstract

Inverse problems are prevalent in numerous scientific and engineering disciplines, where the objective is to determine unknown parameters within a physical system using indirect measurements or observations. The inherent challenge lies in deducing the most probable parameter values that align with the collected data. This study introduces an algorithm for reconstructing parameters by addressing an inverse problem formulated through differential equations underpinned by uncertain boundary conditions or variant parameters. We adopt a Bayesian approach for parameter inference, delineating the establishment of prior, likelihood, and posterior distributions, and the subsequent resolution of the maximum a posteriori problem via numerical optimization techniques. The proposed algorithm is applied to the task of magnetic field reconstruction within a conical domain, demonstrating precise recovery of the true parameter values.

Figures (11)

• Figure 1: Flowchart of our algorithm.
• Figure 2: The physical--computational domain of our problem (left figure) and the finite elements used (right figure). (a) An example of our cone geometry. (b) The finite elements that we use. From left to right: $\mathbb{P}_1$, $\mathbb{P}_2$, and $\mathbb{P}_3$ triangle pyramids.
• Figure 3: Calculation of MF for the unconstrained case (${\nabla ^2}{\bf{B}}({\bf{x}}) = 0$) with analytical BC calculated from Equation \ref{['bound_un_1']} in a 3D cone domain. We plot the magnitude of the MF (a), its vector (b), its divergence (c), and the ratio of the magnitude of the divergence to the magnitude of the MF (d). We can see that $\nabla \cdot {\bf{B}} \sim {10^{ - 3}}$.
• Figure 4: Calculation of MF for the constrained case (where we impose the constraint $\nabla \cdot {\bf{B}} = 0$) with analytical BC calculated from Equation (\ref{['bound_un_1']}) in a 3D cone domain. We plot the magnitude of the MF (a), its vector (b), its divergence (c), and the ratio of the magnitude of the divergence to the magnitude of the MF (d). We can see that $\nabla \cdot {\bf{B}} \sim {10^{ - 5}}$, which is an improvement over what we can see in Figure \ref{['fig:bound_un_1']}.
• Figure 5: Synthetic data of the x-component of the MF to be used in the data-driven reconstruction calculation, as calculated from the unconstrained case forward model with the BCs from Equation (\ref{['bound_un_2']}). In addition, in each sub-figure, a different percentage of the data has been randomly removed to simulate a case of "reconstruction from sparse data". (a) $75\%$ of the data removed. (b) $90\%$ of the data removed. (c) $95\%$ of the data removed. (d) $99\%$ of the data removed.