We define a novel type of ensemble graph convolutional network (GCN) model. Using optimized linear projection operators to map between spatial scales of graph, this ensemble model learns to aggregate information from each scale for its final prediction. We calculate these linear projection operators as the infima of an objective function relating the structure matrices used for each GCN. Equipped with these projections, our model (a Graph Prolongation-Convolutional Network) outperforms other GCN ensemble models at predicting the potential energy of monomer subunits in a coarse-grained mechanochemical simulation of microtubule bending. We demonstrate these performance gains by measuring an estimate of the Floating Point OPerations spent to train each model, 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 of coarse vs. fine training. We examine several such schedules adapted from the algebraic multigrid literature, and quantify the computational benefit of each. We also compare this model to another model which features an optimized coarsening of the input graph. Finally, we derive backpropagation rules for the input of our network model with respect to its output, and discuss how our method may be extended to very large graphs.

我们定义了一种新型的集成图卷积网络（GCN）模型。使用优化的线性投影算子在图形的空间比例之间进行映射，此集成模型将学习汇总每个比例的信息以进行最终预测。我们将这些线性投影算符计算为与每个GCN所使用的结构矩阵相关的目标函数的信息量。有了这些预测，在预测微管弯曲的粗粒度机械化学模拟中，我们的模型（图延长-卷积网络）在预测单体亚基的势能方面优于其他GCN集成模型。我们通过测量用于训练每个模型的浮点运算的估计以及挂钟时间来证明这些性能提升。由于我们的模型具有多种学习能力，可以根据粗训练与细训练的预定时间表在每个等级进行训练。我们研究了从代数多重网格文献改编的几种此类计划，并量化了每种计划的计算效益。我们还将这个模型与另一个模型进行比较，该模型的特征是对输入图进行了优化的粗化。最后，我们针对网络模型的输入导出相对于其输出的反向传播规则，并讨论如何将我们的方法扩展到非常大的图。