  The classical development of neural networks has been primarily for mappings
between a finite-dimensional Euclidean space and a set of classes, or between
two finite-dimensional Euclidean spaces. The purpose of this work is to
generalize neural networks so that they can learn mappings between
infinite-dimensional spaces (operators). The key innovation in our work is that
a single set of network parameters, within a carefully designed network
architecture, may be used to describe mappings between infinite-dimensional
spaces and between different finite-dimensional approximations of those spaces.
We formulate approximation of the infinite-dimensional mapping by composing
nonlinear activation functions and a class of integral operators. The kernel
integration is computed by message passing on graph networks. This approach has
substantial practical consequences which we will illustrate in the context of
mappings between input data to partial differential equations (PDEs) and their
solutions. In this context, such learned networks can generalize among
different approximation methods for the PDE (such as finite difference or
finite element methods) and among approximations corresponding to different
underlying levels of resolution and discretization. Experiments confirm that
the proposed graph kernel network does have the desired properties and show
competitive performance compared to the state of the art solvers.
