  Active learning of physical systems must commonly respect practical safety
constraints, which restricts the exploration of the design space. Gaussian
Processes (GPs) and their calibrated uncertainty estimations are widely used
for this purpose. In many technical applications the design space is explored
via continuous trajectories, along which the safety needs to be assessed. This
is particularly challenging for strict safety requirements in GP methods, as it
employs computationally expensive Monte-Carlo sampling of high quantiles. We
address these challenges by providing provable safety bounds based on the
adaptively sampled median of the supremum of the posterior GP. Our method
significantly reduces the number of samples required for estimating high safety
probabilities, resulting in faster evaluation without sacrificing accuracy and
exploration speed. The effectiveness of our safe active learning approach is
demonstrated through extensive simulations and validated using a real-world
engine example.
