This is a neat idea, and it's interesting to see how well it handles a chaotic system like the double pendulum. I'd love to see a plot of the analytic Lagrangi

This is a neat idea, and it's interesting to see how well it handles a chaotic system like the double pendulum. I'd love to see a plot of the analytic Lagrangian vs the numerical one over the two parameter space of the double pendulum. How uniform is the approximation? Since the double pendulum isn't ergodic, I'd be curious to see if there's a correlation between probability of occupying a state and precision there. If there were it could be used as a hint of where to look for non-ergodic behavior in experimental systems. Another possible fun thing to do with it: imagine you have a chain hanging from two points in the presence of an uneven mass distribution producing gravity. Since you've got the machinery in place for the calculus of variations more generally than just Lagrangian mechanics, you might be able to get the neural net to produce an estimate of the mass distribution from the shape of the chain, which is a toy example of something that might be usable in mineral exploration.