  We present a new application of deep learning to infer the masses of galaxy
clusters directly from images of the microwave sky. Effectively, this is a
novel approach to determining the scaling relation between a cluster's
Sunyaev-Zel'dovich (SZ) effect signal and mass. The deep learning algorithm
used is mResUNet, which is a modified feed-forward deep learning algorithm that
broadly combines residual learning, convolution layers with different dilation
rates, image regression activation and a U-Net framework. We train and test the
deep learning model using simulated images of the microwave sky that include
signals from the cosmic microwave background (CMB), dusty and radio galaxies,
instrumental noise as well as the cluster's own SZ signal. The simulated
cluster sample covers the mass range 1$\times 10^{14}~\rm M_{\odot}$
$<M_{200\rm c}<$ 8$\times 10^{14}~\rm M_{\odot}$ at $z=0.7$. The trained model
estimates the cluster masses with a 1 $\sigma$ uncertainty $\Delta M/M \leq
0.2$, consistent with the input scatter on the SZ signal of 20%. We verify that
the model works for realistic SZ profiles even when trained on azimuthally
symmetric SZ profiles by using the Magneticum hydrodynamical simulations.
