This paper considers a cell-free massive MIMO (multiple-input multiple-output) system that consists of a large number of geographically distributed access points (APs) simultaneously serving multiple user equipments (UEs) on the same time-frequency resources via coherent joint transmission. The performance of the system is evaluated, with maximum ratio and regularized zero-forcing precoding, in terms of the achievable spectral efficiency (SE) under two optimization objectives for the downlink power allocation problem: sum-SE and proportional fairness. Aiming at a less computationally complex as well as a distributed scalable solution, we train a deep neural network (DNN) to perform approximately the same network-wide power allocation. Instead of training our DNN to mimic the actual optimization procedure, we use a heuristic power allocation based on large-scale fading parameters as the input to the DNN. The heuristic input provides better dynamic range while preserving the ratios among the DNN inputs. This allows the use of a simplified structure for the DNN while achieving higher SEs compared to the heuristic scheme.