We explore the use of graph-structured neural-networks (GNNs) to model spatial processes in which there is no {\em a priori} graphical structure. Similar to {\em finite element analysis}, we assign nodes of a GNN to spatial locations and use a computational process defined on the graph to model the relationship between an initial function defined over a space and a resulting function. The encoding of inputs to node states, the decoding of node states to outputs, as well as the mappings defining the GNN are learned from a training set consisting of data from multiple function pairs. The locations of the nodes in space as well as their connectivity can be adjusted during the training process. This graph-based representational strategy allows the learned input-output relationship to generalize over the size and even topology of the underlying space. We demonstrate this method on a traditional PDE problem, a physical prediction problem from robotics, and a problem of learning to predict scene images from novel viewpoints.