  Reinforcement learning algorithms typically consider discrete-time dynamics,
even though the underlying systems are often continuous in time. In this paper,
we introduce a model-based reinforcement learning algorithm that represents
continuous-time dynamics using nonlinear ordinary differential equations
(ODEs). We capture epistemic uncertainty using well-calibrated probabilistic
models, and use the optimistic principle for exploration. Our regret bounds
surface the importance of the measurement selection strategy(MSS), since in
continuous time we not only must decide how to explore, but also when to
observe the underlying system. Our analysis demonstrates that the regret is
sublinear when modeling ODEs with Gaussian Processes (GP) for common choices of
MSS, such as equidistant sampling. Additionally, we propose an adaptive,
data-dependent, practical MSS that, when combined with GP dynamics, also
achieves sublinear regret with significantly fewer samples. We showcase the
benefits of continuous-time modeling over its discrete-time counterpart, as
well as our proposed adaptive MSS over standard baselines, on several
applications.
