  Data fields sampled on irregularly spaced points arise in many applications
in the sciences and engineering. For regular grids, Convolutional Neural
Networks (CNNs) have been successfully used to gaining benefits from weight
sharing and invariances. We generalize CNNs by introducing methods for data on
unstructured point clouds based on Generalized Moving Least Squares (GMLS).
GMLS is a non-parametric technique for estimating linear bounded functionals
from scattered data, and has recently been used in the literature for solving
partial differential equations. By parameterizing the GMLS estimator, we obtain
learning methods for operators with unstructured stencils. In GMLS-Nets the
necessary calculations are local, readily parallelizable, and the estimator is
supported by a rigorous approximation theory. We show how the framework may be
used for unstructured physical data sets to perform functional regression to
identify associated differential operators and to regress quantities of
interest. The results suggest the architectures to be an attractive foundation
for data-driven model development in scientific machine learning applications.
