  The numerical solution of partial differential equations (PDEs) is difficult,
having led to a century of research so far. Recently, there have been pushes to
build neural--numerical hybrid solvers, which piggy-backs the modern trend
towards fully end-to-end learned systems. Most works so far can only generalize
over a subset of properties to which a generic solver would be faced,
including: resolution, topology, geometry, boundary conditions, domain
discretization regularity, dimensionality, etc. In this work, we build a
solver, satisfying these properties, where all the components are based on
neural message passing, replacing all heuristically designed components in the
computation graph with backprop-optimized neural function approximators. We
show that neural message passing solvers representationally contain some
classical methods, such as finite differences, finite volumes, and WENO
schemes. In order to encourage stability in training autoregressive models, we
put forward a method that is based on the principle of zero-stability, posing
stability as a domain adaptation problem. We validate our method on various
fluid-like flow problems, demonstrating fast, stable, and accurate performance
across different domain topologies, equation parameters, discretizations, etc.,
in 1D and 2D.
