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：作为传统偏微分方程数值解法在边界值约束下的替代方案，研究能高效求解此类问题的神经网络已成为热点。本文利用图神经网络与谱图卷积，为两类与时间无关的偏微分方程设计了通用解算子。我们在有限元求解器生成的模拟数据上训练网络，数据包含多种几何形状与非均匀性。与以往研究不同，我们重点考察训练后的算子对未见过场景的泛化能力。具体而言，我们测试了网络在不同形状网格及不同数量非均匀性解叠加场景下的泛化性能。研究发现，在包含丰富有限元网格变化的数据集上训练是实现所有情形下良好泛化结果的关键。基于此，我们认为图神经网络可学习能够泛化至多种属性范围的解算子，且求解速度远快于通用求解器。我们公开的数据集可供使用与扩展，用于验证此类模型在不同条件下的鲁棒性。