Nonlinear dynamical systems exposed to changing forcing can exhibit catastrophic transitions between alternative and often markedly different states. The phenomenon of critical slowing down (CSD) can be used to anticipate such transitions if caused by a bifurcation and if the change in forcing is slow compared to the internal time scale of the system. However, in many real-world situations, these assumptions are not met and transitions can be triggered because the forcing exceeds a critical rate. For example, given the pace of anthropogenic climate change in comparison to the internal time scales of key Earth system components, such as the polar ice sheets or the Atlantic Meridional Overturning Circulation, such rate-induced tipping poses a severe risk. Moreover, depending on the realisation of random perturbations, some trajectories may transition across an unstable boundary, while others do not, even under the same forcing. CSD-based indicators generally cannot distinguish these cases of noise-induced tipping versus no tipping. This severely limits our ability to assess the risks of tipping, and to predict individual trajectories. To address this, we make a first attempt to develop a deep learning framework to predict transition probabilities of dynamical systems ahead of rate-induced transitions. Our method issues early warnings, as demonstrated on three prototypical systems for rate-induced tipping, subjected to time-varying equilibrium drift and noise perturbations. Exploiting explainable artificial intelligence methods, our framework captures the fingerprints necessary for early detection of rate-induced tipping, even in cases of long lead times. Our findings demonstrate the predictability of rate-induced and noise-induced tipping, advancing our ability to determine safe operating spaces for a broader class of dynamical systems than possible so far.