As an alternative to classical numerical solvers for partial differential equations (PDEs) subject to boundary value constraints, there has been a surge of interest in investigating neural networks that can solve such problems efficiently. In this work, we design a general solution operator for two different time-independent PDEs using graph neural networks (GNNs) and spectral graph convolutions. We train the networks on simulated data from a finite elements solver on a variety of shapes and inhomogeneities. In contrast to previous works, we focus on the ability of the trained operator to generalize to previously unseen scenarios. Specifically, we test generalization to meshes with different shapes and superposition of solutions for a different number of inhomogeneities. We find that training on a diverse dataset with lots of variation in the finite element meshes is a key ingredient for achieving good generalization results in all cases. With this, we believe that GNNs can be used to learn solution operators that generalize over a range of properties and produce solutions much faster than a generic solver. Our dataset, which we make publicly available, can be used and extended to verify the robustness of these models under varying conditions.
翻译:作为受边界值限制的局部差异方程式(PDEs)传统数字解答器的替代方案,人们对调查能有效解决这些问题的神经网络的兴趣激增。在这项工作中,我们设计了一个通用的解决方案操作器,用于使用图形神经网络(GNNs)和光谱图相形变异的两种不同时间独立的 PDEs 。我们用有限的元素解析器对网络进行各种形状和异质的模拟数据培训。与以前的工作不同,我们侧重于受过训练的操作器对以往看不见的情景进行概括化的能力。具体地说,我们用不同形状测试模棱两面的神经网络和对不同种类的解决方案的叠加定位。我们发现,在有限的元素 meshes 中,关于多种变异的多种数据集的培训是在所有情况下取得良好概括结果的一个关键要素。我们相信,GNNE可以用来学习解决办法操作器,对各种特性进行普及,并产生比一般解析器快得多的解决方案。我们公开提供的这些模型可以在这些变的模型下使用并扩展这些模型。