  Observations of bright protoplanetary disks often show annular gaps in their
dust emission. One interpretation of these gaps is disk-planet interaction. If
so, fitting models of planetary gaps to observed protoplanetary disk gaps can
reveal the presence of hidden planets. However, future surveys are expected to
produce an ever-increasing number of protoplanetary disks with gaps. In this
case, performing a customized fitting for each target becomes impractical owing
to the complexity of disk-planet interaction. To this end, we introduce DPNNet
(Disk Planet Neural Network), an efficient model of planetary gaps by
exploiting the power of machine learning. We train a deep neural network with a
large number of dusty disk-planet hydrodynamic simulations across a range of
planet masses, disk temperatures, disk viscosities, disk surface density
profiles, particle Stokes numbers, and dust abundances. The network can then be
deployed to extract the planet mass for a given gap morphology. In this work,
first in a series, we focus on the basic concepts of our machine learning
framework. We demonstrate its utility by applying it to the dust gaps observed
in the protoplanetary disk around HL Tau at $10$ au, $30$ au, and $80$ au. Our
network predict planet masses of $80 \, M_{\rm Earth}$, $63 \, M_{\rm Earth}$,
and $70 \, M_{\rm Earth}$, respectively, which are comparable to other studies
based on specialized simulations. We discuss the key advantages of our DPNNet
in its flexibility to incorporate new physics, any number of parameters and
predictions, and its potential to ultimately replace hydrodynamical simulations
for disk observers and modelers.
