  We demonstrate the ability of convolutional neural networks (CNNs) to
mitigate systematics in the virial scaling relation and produce dynamical mass
estimates of galaxy clusters with remarkably low bias and scatter. We present
two models, CNN$_\mathrm{1D}$ and CNN$_\mathrm{2D}$, which leverage this deep
learning tool to infer cluster masses from distributions of member galaxy
dynamics. Our first model, CNN$_\text{1D}$, infers cluster mass directly from
the distribution of member galaxy line-of-sight velocities. Our second model,
CNN$_\text{2D}$, extends the input space of CNN$_\text{1D}$ to learn on the
joint distribution of galaxy line-of-sight velocities and projected radial
distances. We train each model as a regression over cluster mass using a
labeled catalog of realistic mock cluster observations generated from the
MultiDark simulation and UniverseMachine catalog. We then evaluate the
performance of each model on an independent set of mock observations selected
from the same simulated catalog. The CNN models produce cluster mass
predictions with lognormal residuals of scatter as low as $0.132$ dex, greater
than a factor of 2 improvement over the classical $M$-$\sigma$ power-law
estimator. Furthermore, the CNN model reduces prediction scatter relative to
similar machine learning approaches by up to $17\%$ while executing in
drastically shorter training and evaluation times (by a factor of 30) and
producing considerably more robust mass predictions (improving prediction
stability under variations in galaxy sampling rate by $30\%$).
