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

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° azimuthal ambiguity assuming a single layer, either ignoring the intrinsic nonplanar physical geometry of constant optical-depth surfaces (e.g., the Wilson depression in sunspots) or correcting the effect as a postprocessing 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 height 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.