The inverse dynamics problem for a robotic manipulator is to compute the torques needed at the joints to drive it along a given trajectory; it is beneficial to be able to learn this function for adaptive control. A given robot manipulator will often need to be controlled while holding different loads in its end effector, giving rise to a multi-task learning problem. We show how the structure of the inverse dynamics problem gives rise to a multi-task Gaussian process prior over functions, where the inter-task similarity depends on the underlying dynamic parameters. Experiments demonstrate that this multi-task formulation generally improves performance over either learning only on single tasks or pooling the data over all tasks.