以有限差分和有限元为代表的数值方法在计算电磁学等领域有广泛应用，近年来，深度学习等数据驱动的方法在各个领域取得了巨大的成功，并开始应用于计算电磁学领域。利用神经网络加速传统数值算法的研究不仅有诸多实际应用，同时也在不断地拓展其理论体系。在深度学习方法中，将人们已熟知的物理规律同神经网络方法相结合的物理融合神经网络（Physics-informed Neural Network，PINNs）展现出了较好的数值精度和泛化能力。本文系统地探讨了物理融合神经网络在电磁计算运用中的具体形式和技巧，并就计算电磁的典型场景为例，探究并拓展该方法的可行性，并提高计算效率。本文创新点总结如下：1) 对电磁问题的多种数学形式进行了分析，提出了改善网络训练效率的策略，通过应用变分原理设计目标函数，降低了静电场问题的微分阶数，提升了求解静电场问题的计算精度和计算效率。2) 针对物理融合神经网络中奇异点难于描述的问题，提出了基于变分原理等方法的 PINNs 计算框架，规避了源的奇异性，提高了计算的稳定性。3) 利用物理融合神经网络形式灵活的特点，比较了时域问题中边界条件省略、变计算区域以及传统的完全匹配层（Perfect Matched Layer，PML）等不同的边界条件处理方法，数值实验结果表明，这些方法在不同场景下均有其适用性和优势，为实际应用提供了参考依据。
Numerical methods, represented by finite difference and finite element methods, havebeen widely used in areas such as computational electromagnetics. In recent years, datadriven methods such as deep learning have achieved tremendous success in various fieldsand have begun to be applied in computational electromagnetics. Research on using neuralnetworks to accelerate traditional numerical algorithms has not only numerous practicalapplications but also continuously expands its theoretical system.Among deep learning methods, Physics-informed Neural Networks (PINNs), whichcombine well-known physical laws with neural network methods, have shown good numerical accuracy and generalization capabilities. This thesis systematically explores thespecific forms and techniques of physics-informed neural networks in electromagneticcomputations and investigates and expands the feasibility of this method, taking typicalcomputational electromagnetics scenarios as examples, and improving computational efficiency. The innovations of this thesis are summarized as follows:1) We analyze various mathematical forms of electromagnetic problems and proposestrategies to improve network training efficiency. By applying variational principlesto design objective functions, we reduce the order of differentiation in electrostaticfield problems, thus enhancing the computational accuracy and efficiency of solvingelectrostatic field problems.2) To address the difficulty of describing singular points in physics-informed neuralnetworks, we propose a PINNs computational framework based on variational principles and other methods, which circumvents the singularity of sources and improves computational stability.3) By leveraging the flexibility of the physical-informed neural network form, differentboundary condition handling methods, such as missing boundary conditions, variable computational domains, and traditional perfect matched layers (PMLs), arecompared in the context of time-domain problems. Numerical experimental resultsshow that these methods have their own applicability and advantages in differentscenarios, providing a reference for practical applications.
