Maximizing oil production from gas-lifted oil wells entails solving Mixed-Integer Linear Programs (MILPs). As the parameters of the wells, such as the basic-sediment-to-water ratio and the gas-oil ratio, are updated, the problems must be solved repeatedly. Instead of relying on costly exact methods or the accuracy of approximate methods, in this paper, we train deep-learning-based heuristics that provide values for all integer variables based on the parameters of the wells, early fixing their values in the original problem, which becomes a linear program. We propose two approaches for developing such a heuristic: a supervised learning approach, which requires the optimal integer values for several instances of the original problem to train a deep learning model, and a weakly-supervised learning approach, which requires only solutions for the early-fixed linear problem with random integer values. Our results show that our early-fixing heuristics reduce the runtime in 71.11%, and that the supervised learning model can correctly guess the optimal integer values 99.78% of the time. Furthermore, the weakly-supervised learning model can provide significant values for early fixing, despite never seeing the optimal values during training.