We are developing a self-learning mechatronic golf robot using combined data-driven and physics-based methods, to have the robot autonomously learn to putt the ball from an arbitrary point on the green. Apart from the mechatronic control design of the robot, this task is accomplished by a camera system with image recognition and a neural network for predicting the stroke velocity vector required for a successful hole-in-one. To minimize the number of time-consuming interactions with the real system, the neural network is pretrained by evaluating basic physical laws on a model, which approximates the golf ball dynamics on the green surface in a data-driven manner. Thus, we demonstrate the synergetic combination of data-driven and physics-based methods on the golf robot as a mechatronic example system.