This is a really interesting idea, using machine learning to analyze dynamical systems. I've had this idea kicking around in my head about using machine learning on complex time series data for a sort of anomaly detection. You could use example time series data to train a neural network to find conserved quantities in the system, essentially trying to learn the internal energy of the system, and then changes in that internal energy would be your anomalies.

It looks like this paper is getting at the same basic idea, though obviously actually fleshed out and with better results than I would have thought were possible. It's going not for anomaly detection but actual forecasting, and directly isolating the non-conserved force.

I'll need to start reading the related papers, I wonder if people have tried doing this with nonlinear equations, or systems where the dynamics aren't well defined? I imagine actually forecasting a nonlinear system wouldn't be possible beyond short time frames, but maybe you could still learn the underlying equation and dynamics? It would be interesting to see if you could give a neural network a bunch of nonlinear data and have it find equilibrium points, identify regions where the dynamics are almost linear, etc.