论文标题
GCN-FFNN：一种用于学习偏微分方程的学习解决方案的两潮深模型
GCN-FFNN: A Two-Stream Deep Model for Learning Solution to Partial Differential Equations
论文作者
论文摘要
本文介绍了基于图形卷积网络（GCN）体系结构和前馈神经网络（FFNN）的新型两流深模型，用于学习非线性偏微分方程（PDES）的解决方案。该模型旨在分别使用与GCN和FFNN模型相对应的两个流同时合并图形和网格输入表示。每个流层都会接收并处理其自己的输入表示。与接收类似网格的结构的FFNN相反，GCN流层在图输入数据上运行，其中通过图的邻接矩阵合并了邻域信息。这样，提出的GCN-FFNN模型从两种类型的输入表示，即通过PDE域的离散化获得的两种类型的输入表示。 GCN-FFNN模型分为两个阶段。在第一阶段，每个流的模型参数是分别训练的。这两种流都采用相同的误差函数来调整其参数，以实施模型以满足给定的PDE及其在网格或图形置（训练）数据上的初始和边界条件。在第二阶段，两流层的学习参数被冷冻，并且其学习的表示解被馈送到完全连接的图层，其参数使用先前使用的误差函数学习。在PDE域内和外部的测试数据上测试了学习的GCN-FFNN模型。获得的数值结果证明了对单个GCN和FFNN模型的适用性和效率，对1d-Burgers，1d-Schrödinger，2d-Burgers和2d-Schrödinger方程的适用性和效率。
This paper introduces a novel two-stream deep model based on graph convolutional network (GCN) architecture and feed-forward neural networks (FFNN) for learning the solution of nonlinear partial differential equations (PDEs). The model aims at incorporating both graph and grid input representations using two streams corresponding to GCN and FFNN models, respectively. Each stream layer receives and processes its own input representation. As opposed to FFNN which receives a grid-like structure, the GCN stream layer operates on graph input data where the neighborhood information is incorporated through the adjacency matrix of the graph. In this way, the proposed GCN-FFNN model learns from two types of input representations, i.e. grid and graph data, obtained via the discretization of the PDE domain. The GCN-FFNN model is trained in two phases. In the first phase, the model parameters of each stream are trained separately. Both streams employ the same error function to adjust their parameters by enforcing the models to satisfy the given PDE as well as its initial and boundary conditions on grid or graph collocation (training) data. In the second phase, the learned parameters of two-stream layers are frozen and their learned representation solutions are fed to fully connected layers whose parameters are learned using the previously used error function. The learned GCN-FFNN model is tested on test data located both inside and outside the PDE domain. The obtained numerical results demonstrate the applicability and efficiency of the proposed GCN-FFNN model over individual GCN and FFNN models on 1D-Burgers, 1D-Schrödinger, 2D-Burgers and 2D-Schrödinger equations.
