Utilizing machine learning to address partial differential equations (PDEs) presents significant challenges due to the diversity of spatial domains and their corresponding state configurations, which complicates the task of encompassing all potential scenarios through data-driven methodologies alone. Moreover, there are legitimate concerns regarding the generalization and reliability of such approaches, as they often overlook inherent physical constraints. In response to these challenges, this study introduces a novel machine-learning architecture that is highly generalizable and adheres to conservation laws and physical symmetries, thereby ensuring greater reliability. The foundation of this architecture is graph neural networks (GNNs), which are adept at accommodating a variety of shapes and forms. Additionally, we explore the parallels between GNNs and traditional numerical solvers, facilitating a seamless integration of conservative principles and symmetries into machine learning models. Our findings from experiments demonstrate that the model's inclusion of physical laws significantly enhances its generalizability, i.e., no significant accuracy degradation for unseen spatial domains while other models degrade. The code is available at https://github.com/yellowshippo/fluxgnn-icml2024.

翻译：利用机器学习解决偏微分方程（PDE）面临重大挑战，这主要源于空间域及其对应状态配置的多样性，仅通过数据驱动方法难以涵盖所有潜在场景。此外，此类方法常忽略固有的物理约束，其泛化能力与可靠性亦存在合理质疑。为应对这些挑战，本研究提出一种新颖的机器学习架构，该架构具有高度泛化性，并遵循守恒定律与物理对称性，从而确保更高的可靠性。该架构以图神经网络（GNNs）为基础，其擅长适应各种形状与形式。此外，我们探讨了GNN与传统数值求解器之间的相似性，促进了守恒原理与对称性在机器学习模型中的无缝集成。实验结果表明，模型对物理定律的纳入显著增强了其泛化能力，即在未见过的空间域上未出现显著的精度下降，而其他模型则表现退化。代码可在 https://github.com/yellowshippo/fluxgnn-icml2024 获取。