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In addition to requiring the accuracy and computational efficiency for solving low-frequency subsurface sensing problem on the airborne transient electromagnetics (ATEMs), to the best of our knowledge, the complexity of subsurface sensing problems should also be considered in order to reduce more and more computational resources, particularly for a large-scale complicated multis-cale problem with a difference between background and targets. For simulating the open-domain, the finite-thickness perfectly matched layer is used to truncate the computational region, while the whole domain becomes larger so that the problem turns more complex. As a result, we propose a novel perfectly matched monolayer (PMM) model based on the extreme gradient boosting (XGB), which is selected and added to further improve the performance during the finite-difference time-domain (FDTD) simulation. The proposed XGB-based PMM model can achieve higher accuracy by using the ensemble learning method of feature attention, and has less memory and time consumption at the same time. Besides, this model has significant advantages in terms of model training stability and its lightweight due to the fact that it relies on the characteristics of traditional machine learning models. Finally, three-dimensional numerical simulations of ATEM problems are carried out to prove the validity and stability of the proposal. The proposed model can not only achieve advantages in numerical accuracy, efficiency and problem complexity, but also be integrated into the FDTD solver to deal with the low-frequency ATEM problems.
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Keywords:
- extreme gradient boosting /
- perfectly matched monolayer /
- machine learning /
- finite-difference time-domain
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Google Scholar
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图 1 XGB的训练过程
Figure 1. Training process of XGB.
图 2 三维ATEM问题的几何模型
Figure 2. Geometry of 3D ATEM Problem.
图 3 训练过程的精度 (a)树深度; (b)叶子的最小样本分割数; (c)单个样本的抽样比
Figure 3. Accuracies of training process: (a) Tree depth; (b) minimum number of sample splits for the leaves; (c) sampling ratio for a single sample.
图 4 训练过程中Ex, Ey, Hz的相对误差
Figure 4. Relative errors of Ex, Ey, and Hz during the training process.
图 5 三维ATEM问题的几何模型
Figure 5. Geometry of 3D ATEM problems.
图 6 双极方波脉冲, 持续时间为0.04 s, 上升时间为0.0025 s, 下降时间为0.0125 s
Figure 6. Bipolar square wave pulse with duration of 0.04 s, rise time 0.0025 s, and fall time 0.0025 s.
图 7 对于案例A, 利用五种不同方法所取得的二次磁场 (a)时间轴上二次场数值解与预测解对比; (b)相对反射误差计算
Figure 7. Secondary Hz achieved by five different methods for case A: (a) Comparison of secondary Hz field between numerical methods and machine learning methods; (b) computation of relative reflection errors.
图 8 对于案例B, 利用五种不同方法所取得的二次磁场 (a)时间轴上二次场数值解与预测解对比; (b)相对反射误差计算
Figure 8. Secondary Hz achieved by five different methods for case B: (a) Comparison of secondary Hz field between numerical methods and machine learning methods; (b) computation of relative reflection errors.
表 1 基于XGB的ABC模型算法
Table 1. XGB-based ABC algorithms.
算法 基于XGB的ABC模型 输入 训练集、验证集、测试集 初始化 初始化第一颗决策树$ {T_1} $, 并基于目标函数最小来计算决策树权重: $ \min O = -\displaystyle \frac{1}{2}\sum\limits_{j = 1}^T {\frac{G}{{H + \lambda }}} + \gamma T $, $ {w_j} = - \displaystyle\frac{{{G_j}}}{{{H_j} + \lambda }} $ 初始化总模型并将第一棵数加入$ {F_1}(x) = {T_1} $ 初始化残差变量空间$ M $ 循环 计算总模型残差: $ L =\displaystyle \sum\limits_{j = 1}^T {\left[ {{G_j}{x_j} + \frac{1}{2}({H_j} + \lambda )x_j^2} \right]} + \gamma T $ 基于残差数据构建新的决策树$ {T_i} $: $ \min O = - \displaystyle\frac{1}{2}\sum\limits_{j = 1}^T {\frac{G}{{H + \lambda }}} + \gamma T $, $ {w_j} = -\displaystyle \frac{{{G_j}}}{{{H_j} + \lambda }} $ 更新总模型: $ {F_{i + 1}}(x) = {F_i}(x) + {T_i}(x) $ 在验证集中测试模型 停止 收敛或达到预设迭代次数 
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表 2 以0.04 s为周期的计算时间
Table 2. Computational time for 0.04 s as a period
不同实现方法 CPU 所耗
时间/s具有10层PML截断的传统FDTD方法 9711.21 具有5层CFS-PML截断的隐式FDTD方法 342.06 内嵌复频率因子的深度可微森林
完全匹配单层模型115.21 内嵌复频率因子的深度决策树神经
图灵机模型114.51 内嵌双极因子的梯度增强决策树
完全匹配单层模型112.33
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[1] Liu H, Zhao X, Yu Y, Qiu S, Ji Y 2023 IEEE Trans. Geosci. Remote Sens. 61 2002114
Google Scholar
[2] Feng N, Zhang Y, Sun Q, Zhu J, Joines W, Liu Q 2018 IEEE Trans. Antennas Propag. 66 2967
Google Scholar
[3] Qi Y Z, Huang L, Zhangh J G, Fang G Y 2013 Acta Phys. Sin. 62 234201 [齐有政, 黄玲, 张建国, 方广有 2013 物理学报 62 234201]
Google Scholar
Qi Y Z, Huang L, Zhangh J G, Fang G Y 2013 Acta Phys. Sin.62 234201 Google Scholar
[4] Chen J, Li J, Liu Q 2017 IEEE Trans. Antennas Propag. 65 1896
Google Scholar
[5] Mou C H, Chen J, Fan K H, Lu Y 2022 Acta Phys. Sin. 71 184101 [牟春晖, 陈娟, 范凯航, 鲁艺 2022 物理学报 71 184101]
Google Scholar
Mou C H, Chen J, Fan K H, Lu Y 2022 Acta Phys. Sin.71 184101 Google Scholar
[6] Xie G D, Hou G L, Niu K K, Feng N X, Fang, M, Li Y S, Huang Z X 2023 Acta Phys. Sin. 72 150201 [谢国大, 侯桂林, 牛凯坤, 冯乃星, 方明, 李迎松, 黄志祥 2023 物理学报 72 150201]
Google Scholar
Xie G D, Hou G L, Niu K K, Feng N X, Fang, M, Li Y S, Huang Z X 2023 Acta Phys. Sin.72 150201 Google Scholar
[7] Chen J 2018 J. Comput. Phys. 363 256
Google Scholar
[8] Wang F, Wei B, Li L Q 2014 Acta Phys. Sin. 63 104101 [王飞, 魏兵, 李林茜 2014 物理学报 63 104101]
Google Scholar
Wang F, Wei B, Li L Q 2014 Acta Phys. Sin.63 104101 Google Scholar
[9] Wang J, Yin W 2013 IEEE Trans. Antennas Propag. 61 299
Google Scholar
[10] Zhu X M, Ren X C, Guo L X 2014 Acta Phys. Sin. 63 054101 [朱小敏, 任新成, 郭立新 2014 物理学报 63 054101]
Google Scholar
Zhu X M, Ren X C, Guo L X 2014 Acta Phys. Sin.63 054101 Google Scholar
[11] Zhang Y, Feng N, Wang L, Guan Z, Liu Q 2020 IEEE Trans. Antennas Propag. 68 366
Google Scholar
[12] Zhan Q, Zhuang M, Sun Q, Ren Q, Ren Y, Mao Y, Liu Q 2017 IEEE Trans. Geosci. Remote Sens. 55 5577
Google Scholar
[13] Fang Y, Xi X, Wu J, Liu J, Pu Y 2016 IEEE Trans. Microwave Theory Tech. 64 1957
Google Scholar
[14] Fang Y, Xi X, Liu J, Pu Y, Zhao Y, Luo R 2018 IEEE Trans. Antennas Propag. 66 6209
Google Scholar
[15] Feng N, Yue Y, Liu Q 2015 IEEE Trans. Microwave Theory Tech. 63 877
Google Scholar
[16] Chen J, Wang J 2009 IEEE Trans. Antennas Propag. 57 3375
Google Scholar
[17] Yang S, Chen Z, Yu Y, Yin W 2012 IEEE Trans. Antennas Propag. 60 1995
Google Scholar
[18] Feng N, Zhang Y, Tian X, Zhu J, Joines W, Wang G 2019 IEEE Trans. Microwave Theory Tech. 67 3260
Google Scholar
[19] Feng N, Zhang Y, Wang G 2022 IEEE Trans. Microwave Theory Tech. 70 1026
Google Scholar
[20] Bishop C 2006 Pattern Recognition and Machine Learning (Springer) pp1–674
[21] Cai Y, Huang Y, Feng N, Huang Z 2023 IEEE Trans. Microwave Theory Tech. 71 3284
Google Scholar
[22] Feng N, Chen Y, Hong B, Huang Z 2023 IEEE Trans. Microwave Theory Tech. 71 3294
Google Scholar
[23] Li L, Ruan H, Liu C, Li Y, Shuang Y, Alù A, Qiu C, Cui T 2019 Nat. Commun. 10 1082
Google Scholar
[24] Pérez-López D, López A, DasMahapatra P, Capmany J 2020 Nat. Commun. 11 6359
Google Scholar
[25] Kim K, Ha I, Kim M, Choi J, Won P, Jo S, Ko S 2020 Nat. Commun. 11 2149
Google Scholar
[26] Tang Z, Li S, Xu J, Zhang H 2023 Opt. Lett. 48 4416
Google Scholar
[27] Tang Z, Xu J, Wang S, Zhang H 2023 Diamond Relat. Mater. 137 110091
Google Scholar
[28] Yao H, Jiang L 2019 IEEE Antennas Wirel. Propag. Lett. 18 192
Google Scholar
[29] Feng N X, Chen Y S, Zhang Y X, Tong M S, Zeng Q S, Wang G P 2021 IEEE Microwave Wirel. Compon. Lett. 31 541
Google Scholar
[30] Chen Y, Zhang Y, Wang H, Feng N, Yang L, Huang Z 2023 IEEE Trans. Electromagn. Compat. DOI: 10.1109/TEMC.2023.3273724
[31] Feng N, Li J 2013 J. Comput. Phys. 232 318
Google Scholar
[32] Feng N, Yue Y, Zhu C, Wan L, Liu Q 2015 J. Comput. Phys. 285 71
Google Scholar
[33] Berenger J 1994 J. Comput. Phys. 114 185
Google Scholar
[34] Chew W, Weedon W 1994 Microwave Opt. Technol. Lett. 7 599
Google Scholar
[35] Kuzuoglu M, Mittra R 1996 IEEE Microwave Wirel. Compon. Lett. 6 447
Google Scholar
[36] Roden J, Gedney S 2000 Microwave Opt. Technol. Lett. 27 334
Google Scholar
[37] Becacha E, Petropoulus P, Gedney S 2004 IEEE Trans. Antennas Propag. 52 1335
Google Scholar
[38] Berenger J 2002 IEEE Trans. Antennas Propag. 50 258
Google Scholar
[39] Correia D, Jin J 2006 Microwave Opt. Technol. Lett. 48 2121
Google Scholar
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