Spectropolarimetric Inversion in Four Dimensions with Deep Learning (SPIn4D): II. A Physics-Informed Machine Learning Method for 3D Solar Photosphere Reconstruction

TL;DR

SPIn4D introduces a physics-informed, 3D UNet-based approach (HDD) to reconstruct the lower solar atmosphere from multi-height spectropolarimetric inversions by jointly resolving azimuthal ambiguity and geometric height while enforcing a divergence-free magnetic field, $\nabla\cdot\mathbf{B}=0$. The method predicts the disambiguated vector field $\mathbf{B}$ and the height field $Z$ on a deformable grid, using a suite of losses including $loss_{div}$, $loss_{div,0}$, $loss_{smooth}$, and $loss_{mon}$, and a post-processing step to finalize azimuth orientation. Tests on MURaM simulations of quiet Sun, plage, and sunspot show high azimuth-recovery in strong-field regions, accurate geometric heights with correlations up to $R\approx 0.98$, and reliable 3D distributions of electric current density $\mathbf{J}$ and Lorentz force $\mathbf{F}$. The approach provides a general, adaptable framework for obtaining physically consistent 3D magnetic structures from inversion outputs, with potential applications to DKIST observations and time-series analyses.

Abstract

Inferring the three-dimensional (3D) solar atmospheric structures from observations is a critical task for advancing our understanding of the magnetic fields and electric currents that drive solar activity. In this work, we introduce a novel, Physics-Informed Machine Learning method to reconstruct the 3D structure of the lower solar atmosphere based on the output of optical depth sampled spectropolarimetric inversions, wherein both the fully disambiguated vector magnetic fields and the geometric height associated with each optical depth are returned simultaneously. Traditional techniques typically resolve the 180-degree azimuthal ambiguity assuming a single layer, either ignoring the intrinsic non-planar physical geometry of constant optical-depth surfaces (e.g., the Wilson depression in sunspots), or correcting the effect as a post-processing step. In contrast, our approach simultaneously maps the optical depths to physical heights, and enforces the divergence-free condition for magnetic fields fully in 3D. Tests on magnetohydrodynamic simulations of quiet Sun, plage, and a sunspot demonstrate that our method reliably recovers the horizontal magnetic field orientation in locations with appreciable magnetic field strength. By coupling the resolutions of the azimuthal ambiguity and the geometric heights problems, we achieve a self-consistent reconstruction of the 3D vector magnetic fields and, by extension, the electric current density and Lorentz force. This physics-constrained, label-free training paradigm is a generalizable, physics-anchored framework that extends across solar magnetic environments while improving the understanding of various solar puzzles.

Figures (25)

• Figure 1: An illustration of our task and the mesh grid in our computational domain. Here $x$ and $y$ are image plane axes, and $z$-axis is along the line of sight. The curved mesh presents the depression along the LOS for the optical-depth layers. The example given is NOAA AR 11158 as observed on 2011-02-15 in the continuum by SDO/HMI.
• Figure 2: Panel (a) shows the flowchart for training the model. The inputs are the 3D magnetic vector fields with azimuthal ambiguity, and an initial guess of the geometric heights associated with each optical-depth layer. The UNet3D$_B$ network predicts 3D vector magnetic field, and the UNet3D$_Z$ network provides the geometric heights. They are combined to compute a custom loss function with physics law encoded (\ref{['eq:loss']}). Panel (b) shows the post-processing step that assembles the outputs from the neural networks (prediction) and the original input magnetic fields into the final output. Panel (c) shows an example of the 3D data, split with overlaps between nearby subfields. These individual subfields are used as the input for training. Panel (d) presents the UNet3D structure, reproduced from Figure 2 in 10.1007/978-3-319-46723-8_49.
• Figure 3: Panel (a) illustrates the mesh cell structure and the convention for vertex indexing. Panel (b) shows the top surfaces of a mesh cell at the center, and its projection to three planes perpendicular to $\hat{\bm{x}}$, $\hat{\bm{y}}$, and $\hat{\bm{z}}$, respectively.
• Figure 4: MURaM simulations used for this study. Panels (a)--(c) present the vertical components of the magnetic field, $B_z$, on the $\log_{10}\tau=-1$ layer, for the quiet Sun, the plage region, and the sunspot runs. Panels (d)--(f) show the subregion tested in \ref{['sec:test']}.
• Figure 5: Disambiguation results for the plage simulation on $\log_{10}\tau=-1$, the same region as \ref{['fig:3']}(e). Columns, from left to right, show disambiguated $B_x$, $B_y$, and azimuth angle $\Theta$. Rows, from top to bottom, present the ground truth, HDD results, and ME0 results, respectively. HDD's $B_x$ and $B_y$ in Panels d and e are the direct outputs of the neural network, whereas $\Theta$ represents the final result after post-processing. Selected regions with "checkerboard" pattern are highlighted with black and white boxes in Panels d and i.