Galaxy clusters are useful laboratories to investigate the evolution of the Universe, and accurate measurement of their total masses allows us to constrain important cosmological parameters. However, estimating mass from observations that use different methods and spectral bands introduces various systematic errors. Here we evaluate the use of a convolutional neural network (CNN) to reliably and accurately infer the masses of galaxy clusters from the Compton-y parameter maps provided by the Planck satellite. The CNN is trained with mock images generated from hydrodynamic simulations of galaxy clusters, with Planck's observational limitations taken into account. We observe that the CNN approach is not subject to the usual observational assumptions, and therefore is not affected by the same biases. By applying the trained CNNs to the real Planck maps, we find cluster masses compatible with Planck measurements within a 15% bias. Finally, we show that this mass bias can be explained by the well-known hydrostatic equilibrium assumption in Planck masses, and the different parameters in the integrated Compton-y signal and the mass scaling laws. This work highlights that CNNs, supported by hydrodynamic simulations, are a promising and independent tool for estimating cluster masses with high accuracy, which can be extended to other surveys as well as to observations in other bands.