Abstract：Data-driven algorithms based on deep neural networks have been widely used for rapid electromagnetic field solutions, but the accuracy of their calculation results is highly dependent on sufficient sample data. In the data-driven mode, on the one hand, the number of samples in the parameter space must be increased to ensure model accuracy, which greatly increases the sampling time. On the other hand, in many physical and engineering scenarios, data science does not combine known theoretical physics knowledge, resulting in information waste, slow model convergence or even non-convergence, which affects the calculation accuracy. In contrast, physical-driven algorithms can solve electromagnetic field equations without annotating data by introducing physical prior knowledge. However, when dealing with multi-medium field problems, the boundary conditions between media require additional loss function constraints, resulting in a decrease in the efficiency of the neural network optimization process. At the same time, the complexity of the field structure leads to a cumbersome modeling process, and the difficulty of constructing interface loss function conditions is further increased, which hinders the in-depth application of physical-driven algorithms in electromagnetic field calculations. As a physics-driven algorithm, the deep Ritz method (DRM) combines the Ritz method with deep learning to construct a loss function based on energy functionals. When it reaches an extreme value, the interface conditions are automatically satisfied in the sense of functionals, thus avoiding the explicit construction of interface losses. The use of a single neural network to solve electromagnetic interface problems avoids the complexity of multi-objective optimization. However, DRM uses the output of a neural network as a test function. Although the interface conditions are theoretically satisfied, due to the smooth fitting characteristics of the neural network test function, it is difficult to accurately characterize the solution with drastic gradient changes in actual calculations, resulting in increased errors near the interface. Therefore, this paper proposes an improved DRM architecture to enhance its ability to capture interface features during training. Considering that neural networks tend to give priority to low-frequency or large-scale features, this paper introduces embedded Fourier features to process network inputs. The input features of the neural network are mapped into many frequency domains in the frequency domain space, thereby accelerating the neural network’s ability to capture high-frequency components and the convergence speed. At the same time, an adaptive activation function is introduced to automatically adjust the slope of the activation function during the neural network training process to improve the network’s ability to capture high-frequency features. In order to use the improved DRM for stability training of electromagnetic fields, this paper adopts a fully connected neural network structure with hard boundaries. This method effectively eliminates the boundary loss in the total loss function. And by introducing randomized low-discrepancy sequences, the randomness of sampling is improved, the risk of overfitting is reduced, and the convergence of neural network training is improved. By testing the cases of electrostatic fields and steady magnetic fields, the improved DRM and DRM are compared with the finite element calculation results respectively to verify the effectiveness of the method. The improved deep Ritz method captures high-frequency features more effectively and improves the fitting accuracy of the interface area. After the same round of training, in the magnetic field example, the error of the improved network is reduced to 1.03%, while the error of the original network is 5.46%; in the electric field example, the error of the improved network is reduced to 1.49%, while the error of the original network is 4.43%.
张宇娇, 张强, 孙宏达, 赵志涛, 黄雄峰. 基于深度里兹法的电磁场计算方法[J]. 电工技术学报, 2025, 40(23): 7462-7474.
Zhang Yujiao, Zhang Qiang, Sun Hongda, Zhao Zhitao, Huang Xiongfeng. Electromagnetic Field Computation Methods Based on the Deep Ritz Method. Transactions of China Electrotechnical Society, 2025, 40(23): 7462-7474.
[1] Kou Jiaqing, Zhang Weiwei.Data-driven modeling for unsteady aerodynamics and aeroelasticity[J]. Progress in Aerospace Sciences, 2021, 125: 100725. [2] Wainberg M, Merico D, Delong A, et al.Deep learning in biomedicine[J]. Nature Biotechnology, 2018, 36(9): 829-838. [3] 魏蓉, 陈锦培, 仲林林. 基于深度算子网络的电磁轨道发射速度趋肤效应的快速计算方法[J]. 电工技术学报, 2025, 40(5): 1344-1354. Wei Rong, Chen Jinpei, Zhong Linlin.A fast computational method for velocity skin effect of electromagnetic rail launch based on deep operator network[J]. Transactions of China Electrotechnical Society, 2025, 40(5): 1344-1354. [4] 仲林林, 吴冰钰, 吴奇. 基于元学习的气体放电等离子体电子Boltzmann方程数值求解[J]. 电工技术学报, 2024, 39(11): 3457-3466. Zhong Linlin, Wu Bingyu, Wu Qi. numerical solution of electron Boltzmann equation in gas discharge plasmas based on meta learning[J]. Transactions of China Electrotechnical Society, 2024, 39(11): 3457-3466. [5] Qi Shutong, Wang Yinpeng, Li Yongzhong, et al.Two-dimensional electromagnetic solver based on deep learning technique[J]. IEEE Journal on Multiscale and Multiphysics Computational Techniques, 2020, 5: 83-88. [6] 仲林林, 刘柯妤. 面向电力巡检图像目标检测的联邦学习激励机制[J]. 电工技术学报, 2024, 39(17): 5434-5449. Zhong Linlin, Liu Keyu.Federated-learning incentive mechanism for object detection in power inspection images[J]. Transactions of China Electrotechnical Society, 2024, 39(17): 5434-5449. [7] 张翼, 朱永利. 图信号与图卷积网络相结合的局部放电模式识别方法[J]. 中国电机工程学报, 2021, 41(18): 6472-6481. Zhang Yi, Zhu Yongli, Liu Gang.A partial discharge pattern recognition method combining graph signal andgraph convolutional network[J]. Proceedings of the CSEE, 2021, 41(18): 6472-6481. [8] 刘传洋, 吴一全, 刘景景. 无人机航拍图像中绝缘子缺陷检测的深度学习方法研究进展[J]. 电工技术学报, 2025, 40(9): 2897-2916. Liu Chuanyang, Wu Yiquan, Liu Jingjing.Research progress of deep learning methods for insulator defect detection in UAV based aerial images[J]. Transactions of China Electrotechnical Society, 2025, 40(9): 2897-2916. [9] 张宇娇, 赵志涛, 徐斌, 等. 基于U-Net卷积神经网络的电磁场快速计算方法[J]. 电工技术学报, 2024, 39(9): 2730-2742. Zhang Yujiao, Zhao Zhitao, Xu Bin, et al.Fast calculation method of electromagnetic field based on U-Net convolutional neural network[J]. Transactions of China Electrotechnical Society, 2024, 39(9): 2730-2742. [10] 金亮, 尹振豪, 刘璐, 等. 基于残差U-Net和自注意力Transformer编码器的磁场预测方法[J]. 电工技术学报, 2024, 39(10): 2937-2952. Jin Liang, Yin Zhenhao, Liu Lu, et al.Magnetic field prediction method based on residual U-Net and self-attention transformer encoder[J]. Transactions of China Electrotechnical Society, 2024, 39(10): 2937-2952. [11] 刘云鹏, 高艺倩, 刘刚, 等. 基于U-Net神经网络的油浸式变压器绕组流-热耦合快速计算[J]. 中国电机工程学报, 2024, 44(7): 2897-2910. Liu Yunpeng, Gao Yiqian, Liu Gang, et al.Fast calculation of flow-thermal coupling model of oil-immersed transformer windings based on U-Net neural network[J]. Proceedings of the CSEE, 2024, 44(7): 2897-2910. [12] Ibrahim I, Silva R, Mohammadi M H, et al.Surrogate models for design and optimization of inverter-fed synchronous motor drives[J]. IEEE Transactions on Magnetics, 2021, 57(6): 8203805. [13] Tahkola M, Mukherjee V, Keränen J.Transient modeling of induction machine using artificial neural network surrogate models[J]. IEEE Transactions on Magnetics, 2022, 58(9): 7402204. [14] Raissi M, Perdikaris P, Karniadakis G E.Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations[J]. Journal of Computational Physics, 2019, 378: 686-707. [15] Sirignano J, Spiliopoulos K.DGM: a deep learning algorithm for solving partial differential equations[J]. Journal of Computational Physics, 2018, 375: 1339-1364. [16] E Weinan, Yu Bing. The deep Ritz method: a deep learning-based numerical algorithm for solving varia-tional problems[J]. Communications in Mathematics and Statistics, 2018, 6(1): 1-12. [17] Linghu Jiale, Dong Hao, Nie Yufeng, et al.Higher-order multi-scale deep Ritz method (HOMS-DRM) and its convergence analysis for solving thermal transfer problems of composite materials[J]. Computational Mechanics, 2025, 75(1): 71-95. [18] Malekzadeh Fard K, Baghestani A M.Free vibration analysis of deep doubly curved open shells using the Ritz method[J]. Aerospace Science and Technology, 2017, 69: 136-148. [19] He Cuiyu, Hu Xiaozhe, Mu Lin.A mesh-free method using piecewise deep neural network for elliptic interface problems[J]. Journal of Computational and Applied Mathematics, 2022, 412: 114358. [20] Wang Zhongjian, Zhang Zhiwen.A mesh-free method for interface problems using the deep learning approach[J]. Journal of Computational Physics, 2020, 400: 108963. [21] Chen Tianping, Chen Hong.Approximation capability to functions of several variables, nonlinear functionals, and operators by radial basis function neural networks[J]. IEEE Transactions on Neural Networks, 1995, 6(4): 904-910. [22] Wu Sidi, Lu Benzhuo.INN: Interfaced neural networks as an accessible meshless approach for solving interface PDE problems[J]. Journal of Computational Physics, 2022, 470: 111588. [23] Ying Jinyong, Liu Jiaxuan, Chen Jiaxin, et al.Multi-scale fusion network: a new deep learning structure for elliptic interface problems[J]. Applied Mathema-tical Modelling, 2023, 114: 252-269. [24] Yao Yanzhong, Guo Jiawei, Gu Tongxiang.A deep learning method for multi-material diffusion problems based on physics-informed neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 2023, 417: 116395. [25] 汤蕴璆, 梁艳萍. 电机电磁场的分析与计算[M]. 北京: 机械工业出版社, 2010, 310-315. [26] Gnanasambandam R, Shen Bo, Chung J, et al. Self-scalable tanh (stan): faster convergence and better generalization in physics-informed neural networks[J]. arXiv preprint arXiv:2204. 12589, 2022. [27] 张宇娇, 孙宏达, 赵志涛, 等. 基于物理信息神经网络的电磁场计算方法[J]. 电工技术学报, 2024, 39(17): 5251-5261. Zhang Yujiao, Sun Hongda, Zhao Zhitao, et al.Electromagnetic field calculation method based on physical informed neural network[J]. Transactions of China Electrotechnical Society, 2024, 39(17): 5251-5261. [28] Sukumar N, Srivastava A.Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks[J]. Computer Methods in Applied Mechanics and Engineering, 2022, 389: 114333. [29] Kingma D P, Ba J, Hammad M M. Adam: a method for stochastic optimization[J]. arXiv preprint arXiv:1412. 6980, 2014. [30] 王洪友, 杨光源, 邵可然. 基于无单元伽辽金法的电机电磁场计算[J]. 中国电机工程学报, 2012, 32(15): 148-154, 3. Wang Hongyou, Yang Guangyuan, Shao Keran.Computations for electromagnetic field of synchro-nous generators based on the element-free Galerkin method[J]. Proceedings of the CSEE, 2012, 32(15): 148-154, 3. [31] Owen A B.Scrambling sobol’ and niederreiter-xing points[J]. Journal of Complexity, 1998, 14(4): 466-489. [32] Zeng Xin, Zhang Shuai, Ren Chenhua, et al.Physics informed neural networks for electric field distribu-tion characteristics analysis[J]. Journal of Physics D Applied Physics, 2023, 56(16): 165202.
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