Modelling Jupiter's global and regional magnetic fields using physics-informed neural networks

As is known, neural networks can universally approximate any complex functions. This ground truth naturally makes it a suitable candidate for solution representation of complex partial differential equation (PDE) governed. For planetary magnetic field modelling problem, spherical harmonic functions are most used as standard modelling method. Spherical harmonic method requires globally nearly uniformly distributed observations. Meanwhile this method has quite limited ability for conducting regional field modelling. Instead, neural networks have great potential to deal with global or regional modelling problems. In this work, we thoroughly investigate the representative ability of neural networks for magnetic field modelling problem at global and regional scale, and concentrate on a specific neural network, that is physics-informed neural networks (PINNs) for implementation. PINNs makes it easier to incorporate different kinds of informed physics within a uniform optimization framework. Through synthetic model tests and partial mathematical proof, we showcase the importance of employing natural boundary condition, Laplace equation constraint and Poisson equation constraint at suitable collocation points for a reasonable and accurate magnetic field representation and introduce the detailed scheme for implementation. Finally, we use newly released Juno mission measurements, and present a global PINNs model for Jupiter's magnetic field, and a regional PINNs model for Great Blue Spot (GBS) region. Comparison with spherical harmonic model has been conducted to evaluate the correctness and flexibility of PINNs models.