Probing the solar coronal magnetic field with physics-informed neural networks

While the photospheric magnetic field of our Sun is routinely measured, its extent into the upper atmosphere is typically not accessible by direct observations. Here we present an approach for coronal magnetic-field extrapolation, using a neural network that integrates observational data and the physical force-free magnetic-field model. Our method flexibly finds a trade-off between the observation and force-free magnetic-field assumption, improving the understanding of the connection between the observation and the underlying physics. We utilize meta-learning concepts to simulate the evolution of active region NOAA 11158. Our simulation of 5 days of observations at full cadence (12 minutes) requires less than 12 hours of total computation time, allowing for real-time force-free magnetic-field extrapolations. The application to an analytical magnetic-field solution, a systematic analysis of the time evolution of free magnetic energy and magnetic helicity in the coronal volume, as well as comparison with extreme-ultraviolet observations, demonstrates the validity of our approach. The obtained temporal and spatial depletion of free magnetic energy unambiguously relates to the observed flare activity. The application of physics-informed neural networks enables an estimation of the solar coronal magnetic field in quasi real time. A comparison with extreme-ultraviolet observations reveals that the model provides a realistic approximation and the modelled coronal field has a clear relationship with flaring activity.