At hadron colliders, the net transverse momentum of particles that do not interact with the detector (missing transverse momentum, $\vec{p}_\mathrm{T}^\text{miss}$) is a crucial observable in many analyses. In the standard model, $\vec{p}_\mathrm{T}^\text{miss}$ originates from neutrinos. Many beyond-the-standard-model particles, such as dark matter candidates, are also expected to leave the experimental apparatus undetected. This paper presents a novel $\vec{p}_\mathrm{T}^\text{miss}$ estimator, DeepMET, which is based on deep neural networks that were developed by the CMS Collaboration at the LHC. The DeepMET algorithm produces a weight for each reconstructed particle based on its properties. The estimator is based on the negative vector sum of the weighted transverse momenta of all reconstructed particles in an event. Compared with other estimators currently employed by CMS, DeepMET improves the $\vec{p}_\mathrm{T}^\text{miss}$ resolution by 10$-$30%, shows improvement for a wide range of final states, is easier to train, and is more resilient against the effects of additional proton-proton interactions accompanying the collision of interest.