Solving partial differential equations (PDEs) is an important research means in the fields of physics, biology, and chemistry. As an approximate alternative to numerical methods, PINN has received extensive attention and played an important role in many fields. However, PINN uses a fully connected network as its model, which has limited fitting ability and limited extrapolation ability in both time and space. In this paper, we propose PhyGNNet for solving partial differential equations on the basics of a graph neural network which consists of encoder, processer, and decoder blocks. In particular, we divide the computing area into regular grids, define partial differential operators on the grids, then construct pde loss for the network to optimize to build PhyGNNet model. What's more, we conduct comparative experiments on Burgers equation and heat equation to validate our approach, the results show that our method has better fit ability and extrapolation ability both in time and spatial areas compared with PINN.

翻译：求解偏微分方程是物理、生物和化学领域的重要研究手段。作为数值方法的近似替代方案，PINN受到广泛关注并在多个领域发挥重要作用。然而，PINN采用全连接网络作为其模型，拟合能力有限，且在时间和空间上的外推能力有限。本文在编码器-处理器-解码器架构的图神经网络基础上，提出PhyGNNet用于求解偏微分方程。具体而言，我们将计算区域划分为规则网格，在网格上定义偏微分算子，进而构建偏微分方程损失函数以优化网络，从而建立PhyGNNet模型。此外，我们在Burgers方程和热传导方程上开展对比实验以验证该方法，结果表明，与PINN相比，本方法在时间和空间维度均具有更优的拟合能力和外推能力。