This paper describes applications of two neural networks to improve drift chamber position measurements. One network calculates a data-driven estimate of the drift cell time-to-distance relationship that is conventionally estimated by a numerical calculation based on the anode and cathode wire geometry, wire potentials, and gas properties. The second network additionally uses the full digital waveform of the signal in the drift chamber, hence accessing information on the full ensemble of ionization clusters. This network uses more information than the conventional position estimate that relies exclusively on the arrival time of the first drift electron. In principle, this technique improves resolution even when multiple ionization clusters cannot be separated, in contrast with a cluster-counting technique. The performance of both networks when applied to MEG II drift chamber data is reported and compared to that of a conventional approach.