Interesting, coming from physics rather than an ML background what does it mean "practically" to learn a symmetry of a system? Is it the quantity <=> Noether's theorem being constant?
You want to learn a function that represents the dynamics of your system, either as a function of the system state or some output like a picture of the system. If you just apply some NN technique directly, this is possible but will require a lot of data since the NN doesn't have any knowledge of physics. If you use their system, you are trying to learn the Lagrangian of the system, which contains information on e.g. its symmetries, and bakes in physical knowledge into the learning problem at hand. As a result, less data is needed to learn the system dynamics.
I think more people need to learn to see the positive side of TANSTAAFL...
(I suppose it's learning a symmetry in the following sense: just _what_ is conserved depends on what the Lagrangian is, and so as it's learning the dynamics it's also learning what the energy is that it should be conserving. But at every point in the training process, there's _some_ thing, which we might as well call "energy", which its model conserves.)