> First we deduce formally-exact non-perturbative guiding center equations of motion assuming a hidden symmetry with associated conserved quantity J. We refer to J as the non-perturbative adiabatic invariant.
Simply: this is not just some kind of unsupervised ML black-box magic. There is a formal mathematical solution to something, but it has a certain gap, namely precisely what quantity is conserved and how to calculate it.
> Then we describe a data-driven method for learning J from a dataset of full-orbit α-particle trajectories. [...] Our proposed method for learning J applies on a per-magnetic field basis; changing B requires re-training. This makes it well-suited to stellarator design assessment tasks, such as α-loss fraction uncertainty quantification.
With the formal simplification of the dynamics in hand, the researchers believe that a trained model can then give a useful approximation of the invariant, which allows the formal model, with its unknown parameters now filled in, to be used to model the dynamics.
In a crude way, I think I have a napkin-level sketch of what they're doing here. Suppose we are modeling a projectile, and we know nothing of kinematics. They have determined that the projectile has a parabolic trajectory (the formal part) and then they are using data analysis to find the g coefficient that represents gravitational acceleration (the data-driven part). Obviously, you would never need machine learning in such a very simple case as I have described, but I think it approximates the main idea.