Long-term research questions

[dtch1997]

This is intended more as a long-term store of interesting research questions about neural certificate learning, and I'm not expecting any easy answers. However, I believe that these would be good starting points for future work

1. In the current repository, contraction metrics are jointly learned with the learned controller. Is it instead possible to consider separate learning of a metric and then a controller?
   a. Conceptually, all we need to learn a valid metric is a dataset of trajectory "bundles", and each bundle has the property that every pair in the bundle is getting (qualitatively) closer.
   b. Disentangling the learning this way has the benefit that we no longer need any knowledge of the dynamics notion. The contraction metric loss function only requires $\dot{x}$, which can be estimated directly from historical trajectory data $x(t)$ instead of from an online controller $\dot{x} = f(x, u)$.
   c. A learned contraction metric could be frozen and then used in the normal way during learning of a controller.
   d. Going back to (a), it may be interesting to explore a contrastive setting where we have a dataset of bundles that are either (i) converging or (ii) diverging, and then we can define the appropriate loss function which encourages the metric to be increasing / decreasing respectively.
2. Can the contraction metric guarantee be extended to the partially observable case? This does mean that dynamics become non-Markovian.