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We define a novel type of ensemble Graph Convolutional Network (GCN) model.Using optimized linear projection operators to map between spatial scales ofgraph, this ensemble model learns to aggregate information from each scale forits final prediction. We calculate these linear projection operators as theinfima of an objective function relating the structure matrices used for eachGCN. Equipped with these projections, our model (a GraphProlongation-Convolutional Network) outperforms other GCN ensemble models atpredicting the potential energy of monomer subunits in a coarse-grainedmechanochemical simulation of microtubule bending. We demonstrate theseperformance gains by measuring an estimate of the FLOPs spent to train eachmodel, as well as wall-clock time. Because our model learns at multiple scales,it is possible to train at each scale according to a predetermined schedule ofcoarse vs. fine training. We examine several such schedules adapted from theAlgebraic Multigrid (AMG) literature, and quantify the computational benefit ofeach. We also compare this model to another model which features an optimizedcoarsening of the input graph. Finally, we derive backpropagation rules for theinput of our network model with respect to its output, and discuss how ourmethod may be extended to very large graphs.

Graph Prolongation Convolutional Networks: Explicitly Multiscale Machine Learning on Graphs with Applications to Modeling of Cytoskeleton