Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of
  Non-Euclidean Operators
Blaine Quackenbush Paul J. Atzberger
We introduce Geometric Neural Operators (GNPs) for accounting for geometric
contributions in data-driven deep learning of operators. We show how GNPs can
be used (i) to estimate geometric properties, such as the metric and
curvatures, (ii) to approximate Partial Differential Equations (PDEs) on
manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and
(iv) to solve Bayesian inverse problems for identifying manifold shapes. The
methods allow for handling geometries of general shape including point-cloud
representations. The developed GNPs provide approaches for incorporating the
roles of geometry in data-driven learning of operators.