This paper focuses on the use of a deep learning approach to perform sum-rate-max and max-min power allocation in the uplink of a cell-free massive MIMO network. In particular, we train a deep neural network in order to learn the mapping between a set of input data and the optimal solution of the power allocation strategy. Numerical results show that the presence of the pilot contamination in the cell-free massive MIMO system does not significantly affect the learning capabilities of the neural network, that gives near-optimal performance. Conversely, with the introduction of the shadowing effect in the system the performance obtained with the deep learning approach gets significantly degraded with respect to the optimal one.