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?
Having skimmed the research paper and done some work with both dynamics and ML, my interpretation of their statement is the following:
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.
Yup, but the computer isn't _learning_ it, it's already enforced by the fact that what the computer is learning is a time-independent Lagrangian.
(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.)
Yeah, I was thinking about symmetries <=> conservation laws because of Noether's theorem. Think of regular NN training as not having any symmetries...since they aren't baked into the model. But we can let the model learn symmetries <=> conservation laws by adding the Euler-Lagrange constraint to the forward pass of a NN.
