Computational electromagnetics (CEM) is employed to numerically solve Maxwell's equations, and it has very important and practical applications across a broad range of disciplines, including biomedical engineering, nanophotonics, wireless communications, and electrodynamics. The main limitation of existing CEM methods is that they are computationally demanding. Our work introduces a leap forward in scientific computing and CEM by proposing an original solution of Maxwell's equations that is grounded on graph neural networks (GNNs) and enables the high-performance numerical resolution of these fundamental mathematical expressions. Specifically, we demonstrate that the update equations derived by discretizing Maxwell's partial differential equations can be innately expressed as a two-layer GNN with static and pre-determined edge weights. Given this intuition, a straightforward way to numerically solve Maxwell's equations entails simple message passing between such a GNN's nodes, yielding a significant computational time gain, while preserving the same accuracy as conventional transient CEM methods. Ultimately, our work supports the efficient and precise emulation of electromagnetic wave propagation with GNNs, and more importantly, we anticipate that applying a similar treatment to systems of partial differential equations arising in other scientific disciplines, e.g., computational fluid dynamics, can benefit computational sciences

翻译：计算电磁学（CEM）用于数值求解麦克斯韦方程组，在生物医学工程、纳米光子学、无线通信和电动力学等多个学科领域具有极其重要的实际应用。现有CEM方法的主要局限在于其计算成本高昂。我们的工作通过在科学计算和CEM领域实现一项重大突破，提出了一种基于图神经网络（GNN）的麦克斯韦方程组原创求解方案，从而能够对这些基本数学表达式进行高性能数值求解。具体而言，我们证明了通过离散化麦克斯韦偏微分方程得到的更新方程可以自然地表达为一个具有静态且预确定边权重的两层GNN。基于这一直觉，数值求解麦克斯韦方程组的一种直接方法是在此类GNN的节点之间进行简单的消息传递，从而在保持与传统瞬态CEM方法相同精度的同时，显著节省计算时间。最终，我们的工作支持利用GNN高效且精确地模拟电磁波传播，更重要的是，我们预计将类似处理方法应用于其他科学领域（例如计算流体动力学）中出现的偏微分方程组，将有益于计算科学的发展。