Coronal magnetic-field extrapolation as an inverse problem
Coronal magnetic-field extrapolation means inferring a plausible 3D coronal field from magnetic measurements at the lower boundary, usually a photospheric vector magnetogram.
Plain version: we can measure the magnetic field much better at the solar surface than in the corona, so we use the surface field plus physics assumptions to reconstruct the field above it.
Why the problem exists
The corona is shaped by magnetic fields. Loops, arcades, sigmoids, flares, and CMEs all depend on magnetic energy storage and release.
The problem is observational: the full 3D coronal magnetic field is not routinely measured directly. The usual data product is a lower-boundary magnetogram, such as an SDO/HMI SHARP vector magnetogram.
So the modelling problem is:
Why it is not interpolation
This is not ordinary interpolation. Many different 3D fields can be compatible with similar-looking boundary data.
The field also has to satisfy physics:
- solenoidality: ;
- force balance, often approximated by NLFFF equations;
- reasonable magnetic energy;
- plausible topology and connectivity;
- agreement with the observed lower boundary.
Those constraints turn the task into an inverse problem. We infer the hidden volume from incomplete observations plus a model of how the field should behave.
Why this matters for NF2
NF2 and physics-informed neural coronal extrapolation treats extrapolation as a physics-constrained inverse problem. The neural network is not the physics. It is a flexible way to represent while training against boundary data and equation residuals.
That is why NF2 has to be judged with the same physical diagnostics as other NLFFF methods. A smooth neural field is only useful if it satisfies the coronal magnetic-field problem, not just the optimiser.