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基于多目标PSO混合优化的虚拟样本生成

王丹丹 汤健 夏恒 乔俊飞

王丹丹, 汤健, 夏恒, 乔俊飞. 基于多目标PSO混合优化的虚拟样本生成. 自动化学报, 2022, 45(x): 1−22 doi: 10.16383/j.aas.c211091
引用本文: 王丹丹, 汤健, 夏恒, 乔俊飞. 基于多目标PSO混合优化的虚拟样本生成. 自动化学报, 2022, 45(x): 1−22 doi: 10.16383/j.aas.c211091
Wang Dan-Dan, Tang Jian, Xia Heng, Qiao Jun-Fei. Virtual sample generation method based on hybrid optimization with multi-objective pso. Acta Automatica Sinica, 2022, 45(x): 1−22 doi: 10.16383/j.aas.c211091
Citation: Wang Dan-Dan, Tang Jian, Xia Heng, Qiao Jun-Fei. Virtual sample generation method based on hybrid optimization with multi-objective pso. Acta Automatica Sinica, 2022, 45(x): 1−22 doi: 10.16383/j.aas.c211091

基于多目标PSO混合优化的虚拟样本生成

doi: 10.16383/j.aas.c211091
基金项目: 国家自然科学基金(62073006, 62173120, 62021003), 北京市自然科学基金资助项目(4212032, 4192009)和科技创新2030-“新一代人工智能”重大项目(2021ZD0112301, 2021ZD0112302)资助
详细信息
    作者简介:

    王丹丹:北京工业大学信息学部硕士研究生.主要研究方向为基于虚拟样本生成的小样本数据建模. E-mail. wangdandan@emails.bjut.edu.cn

    汤健:北京工业大学信息学部教授.主要研究方向为小样本数据建模和城市固废处理过程智能控制.本文通信作者. E-mail. freeflytang@bjut.edu.cn

    夏恒:北京工业大学信息学部博士研究生.主要研究方向为小样本数据建模和城市固废焚烧过程二噁英排放预测. E-mail. xiaheng@emails.bjut.edu.cn

    乔俊飞:北京工业大学信息学部教授.主要研究方向为污水处理过程智能控制和神经网络结构设计与优化. E-mail. junfeiq@bjut.edu.cn

Virtual Sample Generation Method Based on Hybrid Optimization With Multi-objective PSO

Funds: Supported by National Natural Science Foundation of China(62073006, 62173120, 62021003), Beijing Natural Science Foundation (4212032, 4192009) and National Key Research and Development Program of China (2021ZD0112301, 2021ZD0112302)
More Information
    Author Bio:

    WANG Dan-Dan Master student at the Faculty of Information Technology, Beijing University of Technology. Her main research interest is small sample modeling based on virtual sample generation

    TANG Jian Professor at the Faculty of Information Technology, Beijing University of Technology. His research interest covers small sample data modeling and intelligent control of municipal solid waste treatment process. Corresponding author of this paper

    XIA Heng Ph.D. candidate at the Faculty of Information Technology, Beijing University of Technology. His research interest covers small sample data modeling and dioxin emission prediction of the municipal solid waste incineration process

    QIAO Jun-Fei Professor at the Faculty of Information Technology, Beijing University of Technology. His research interest covers intelligent control of waste water treatment process and structure design and optimization of neural networks

  • 摘要: 产品质量与污染排放浓度等难测参数的实时检测是实现复杂工业过程优化控制的关键因素之一. 受限于检测技术难度、高时间与经济成本等原因, 难测参数的软测量模型建模样本存在数量少、分布稀疏与不平衡等问题, 严重制约了数据驱动模型的泛化性能. 针对以上问题, 提出一种基于多目标粒子群优化混合优化的虚拟样本生成方法, 首先, 设计综合学习粒子群优化算法的种群表征机制, 使其能够同时编码用于映射模型超参数优化的连续变量和用于虚拟样本选择的离散变量; 然后, 定义具有多阶段多目标特性的综合学习粒子群优化算法适应度函数, 使其能够在确保模型泛化性能的同时最小化虚拟样本数量; 最后, 向虚拟样本生成多目标混合优化任务对综合学习粒子群优化算法进行改进, 使其能够适应虚拟样本优选过程的变维特性并提高优化过程的收敛速度. 同时, 首次借鉴度量学习的指标提出用于评价虚拟样本质量的综合评价指标和分布相似指标. 本文采用混凝土抗压强度和超导临界温度基准数据集验证了所提算法的合理性及有效性, 基于工业数据集构建了面向城市固废焚烧过程的二噁英排放浓度的软测量模型, 进一步验证了所提方法.
  • 图  1  虚拟样本与真实样本间的关系

    Fig.  1  Relationship between virtual samples and real samples

    图  2  基于MOPSO混合优化的虚拟样本生成策略

    Fig.  2  Virtual sample generation strategy based on hybrid optimization with MOPSO

    图  3  基于混合优化策略的粒子设计

    Fig.  3  Particle design based on hybrid optimization strategy

    图  4  非支配解的Pareto前沿

    Fig.  4  Pareto front of nondominant solutions

    图  5  非支配解的建模性能指标对比

    Fig.  5  Comparison of modelling performance indexes of nondominant solutions

    图  7  非支配解的分布相似度对比

    Fig.  7  Comparison of distribution similarity of nondominant solutions

    图  6  非支配解的综合评价指标对比

    Fig.  6  Comparison of comprehensive evaluation indexes of non-dominant solutions

    图  8  基准数据预测输出对比

    Fig.  8  Comparison of prediction output for benchmark data

    图  9  MSWI过程工艺流程图

    Fig.  9  Flow chart of MSWI process

    图  10  非支配解的Pareto前沿-DXN排放浓度

    Fig.  10  Pareto front of nondominated solution-DXN emission concentration

    图  11  非支配解的建模性能、综合评价指标对比

    Fig.  11  Comparison of modeling performance indexes and comprehensive evaluation indexes of nondominant solutions

    表  1  本文所采用的符号的含义

    Table  1  The meaning of the symbols used in this article

    序号 符号 含义
    1 ${\rho ^i}$ 全局最优粒子选择指标
    2 ${\rho _j}$ 虚拟样本综合评价指标
    3 $\eta $ 数据分布相似度
    4 ${{\boldsymbol{F}}}\left( {{\boldsymbol{z}}} \right)$ 多目标优化问题的目标函数集
    5 ${{\boldsymbol{z}}}, z_n^p\left( {t + 1} \right)$ 优化问题决策变量(粒子的位置矢量), 表示第$t + 1$次迭代时, 粒子$p$的第$n$维位置值
    6 ${{\boldsymbol{v}}}, v_n^p\left( {t + 1} \right)$ 粒子的速度矢量, 表示第$t + 1$次迭代时, 粒子$p$的第$n$维速度值
    7 ${w_{{{\rm{inertia}}}}}$ 粒子速度更新的惯性权重
    8 $r_{1n}^p, r_{2n}^p$ 粒子速度更新的随机量, 服从[0, 1]间的均匀分布
    9 ${c_{{{\rm{self}}}}}$ 粒子速度更新的个体学习因子
    10 ${c_{{{\rm{society}}}}}$ 粒子速度更新的社会学习因子
    11 ${{{\boldsymbol{d}}}^p}\left( {t + 1} \right)$ 第$t + 1$次迭代时, 粒子$p$的个体最优
    12 ${{\boldsymbol{g}}}\left( {t + 1} \right)$ 第$t + 1$次迭代时, 种群的全局最优
    13 $E_n^p$ 粒子$p$的第$n$维的学习样例值
    14 ${N_{{{\rm{refresh}}}}}$ 个体最优未更新阈值, 用于控制学习样例的更新
    15 $P_c^p$ 粒子$p$的学习概率, 用于控制学习样例的更新概率
    16 $ran{k^p}$ 粒子$p$个体最优的适应度在种群中排名
    17 $K$ RF模型中决策树数量
    18 ${L_F}$ RF模型中切分特征数
    19 ${\theta _{{{\rm{leaf}}}}}$ RF模型中决策树的叶节点包含样本数量的阈值
    20 $F_{_{{{\rm{sel}}}}}^q$ RF模型中, 决策树的节点$q$最佳切分特征
    21 ${s^q}$ RF模型中, 决策树的节点$q$最佳分裂点取值
    22 $f_{_{{{\rm{tree}}}}}^k\left( \cdot \right)$ RF模型中第$k$个决策树模型
    23 $f_{_{{{\rm{RF}}}}}^{}\left( \cdot \right)$ RF模型
    24 ${{\boldsymbol{z}}}_{_{{{\rm{para}}}}}^{}$ 指导候选虚拟样本生成的参数决策变量
    25 ${{\boldsymbol{z}}}_{_{{{\rm{vss}}}}}^{}$ 筛选候选虚拟样本选择决策变量
    26 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{train}}}}} $ 原始小样本训练集
    27 ${{{\boldsymbol{x}}}_{{{\rm{vsg-min}}}}}, {{{\boldsymbol{x}}}_{{{\rm{vsg-max}}}}}$ 采用改进MTD进行扩展后的输入扩展域的上限和下限
    28 ${y_{{{\rm{vsg-min}}}}}, {y_{{{\rm{vsg-max}}}}}$ 采用改进MTD进行扩展后的输出扩展域的上限和下限
    29 $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{vs-g}}}}} $ 混合插值生成的虚拟样本输入
    30 $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{equal}}}}} , \mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{rand}}}}} $ 等间隔插值、随机插值生成的虚拟样本输入
    31 $\mathop {{\boldsymbol{y}}}\nolimits_{_{{{\rm{vs-g1}}}}} , \mathop {{\boldsymbol{y}}}\nolimits_{_{{{\rm{vs-g2}}}}} $ 基于虚拟样本输入, 结合RF、RWNN映射模型生成的虚拟样本输出
    32 ${{\boldsymbol{R}}}_{{{\rm{vs-g1}}}}^p, {{\boldsymbol{R}}}_{{{\rm{vs-g2}}}}^p$ 基于虚拟样本输入, 结合RF、RWNN映射模型生成的虚拟样本
    33 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs-g}}}}} $ 生成的混合虚拟样本
    34 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs-d}}}}} $ 对$\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs-g}}}}} $进行删减后的候选虚拟样本
    35 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs-s}}}}} $ 对候选虚拟样本进行选择后获得的虚拟样本
    36 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{valid}}}}} $ 原始小样本验证集
    37 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs}}}}} $ 最优虚拟样本
    38 ${f_{{{\rm{num}}}}}({{\boldsymbol{z}}})$ 多目标优化问题的目标之一, 筛选后的虚拟样本数量
    39 ${f_{{{\rm{mod}}}}}({{\boldsymbol{z}}})$ 多目标优化问题的目标之一, 筛选后的虚拟样本与原始训练集构建RF模型的性能指标
    40 $z_{MTD}$ 粒子的参数决策变量之一, 对应基于MTD方法的扩展率${\gamma _{{{\rm{extend}}}}}$
    41 $z_{{{\rm{RF}}}}^{{\rm{1}}}$ 粒子的参数决策变量之一, 对应RF映射模型的分特征数${L_F}$
    42 $z_{{{\rm{RF}}}}^{{\rm{2}}}$ 粒子的参数决策变量之一, 对应RF映射模型中决策树的中叶节点包含样本数量的阈值${\theta _{{{\rm{leaf}}}}}$
    43 ${z_{{{\rm{RWNN}}}}}$ 粒子的参数决策变量之一, 对应RWNN映射模型的隐含层神经元数量$I$
    44 ${\gamma _{{{\rm{extend}}}}}$ 基于MTD方法的扩展率
    45 $I$ RWNN映射模型的隐含层神经元数量
    46 $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{train}}}}} $ 原始小样本训练集输入
    47 ${{{\boldsymbol{y}}}_{{{\rm{train}}}}}$ 原始小样本训练集输出
    48 ${y_{{{\rm{ave}}}}}$ ${{{\boldsymbol{y}}}_{{{\rm{train}}}}}$的均值
    49 ${{{\boldsymbol{y}}}_{{{\rm{high}}}}}, {{{\boldsymbol{y}}}_{{{\rm{low}}}}}$ $\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{train}}}}} $中大于/小于${y_{{{\rm{ave}}}}}$的输出集合
    50 ${y_{{{\rm{max}}}}}, {y_{{{\rm{min}}}}}$ ${{{\boldsymbol{y}}}_{{{\rm{train}}}}}$中最大值、最小值
    51 ${y_{{{\rm{H-ave}}}}}, {y_{{{\rm{L-ave}}}}}$ ${{{\boldsymbol{y}}}_{{{\rm{high}}}}}, {{{\boldsymbol{y}}}_{{{\rm{low}}}}}$的均值
    52 $rat{e_{{{\rm{high}}}}}, rat{e_{{{\rm{low}}}}}$ 域扩展的上下偏度
    53 ${d_{{{\rm{y-high}}}}}, {d_{{{\rm{y-low}}}}}$ ${y_{{{\rm{max}}}}}$和${y_{{{\rm{H-ave}}}}}$间的欧式距离, ${y_{{{\rm{min}}}}}$和${y_{{{\rm{L-ave}}}}}$间的欧式距离
    54 $\mathop N\nolimits_{_{{{\rm{equal}}}}} , \mathop N\nolimits_{_{{{\rm{rand}}}}} $ 等间隔插值、随机插值倍数
    55 ${{\boldsymbol{W}}}, {{\boldsymbol{b}}}$ RWNN模型输入层与隐含层间神经元的连接权重与偏置
    56 ${{\boldsymbol{H}}}_{}^{_{{{\rm{ori}}}}}$ RWNN模型隐含层输出矩阵
    57 ${{\boldsymbol{\beta}}}$ RWNN模型隐含层与输出层神经元的连接权重
    58 $\mathop N\nolimits_{_{{{\rm{vs-g}}}}} \mathop {, N}\nolimits_{_{{{\rm{vs-d}}}}} , \mathop N\nolimits_{_{{{\rm{vs-s}}}}} $ 生成、候选、选择后虚拟样本的数量
    59 ${\theta _{{{\rm{select}}}}}$ 虚拟样本的选择阈值
    60 ${{{\boldsymbol{\tilde z}}}_{{{\rm{vss}}}}}$ 对${{{\boldsymbol{z}}}_{{{\rm{vss}}}}}$进行变维度处理后获得
    61 $F$ 使用虚拟样本集$\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs-s}}}}}$的建模性能指标
    62 $\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{mix}}}}} $ 原始训练集$\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{train}}}}} $与$\mathop {{\boldsymbol{R}}}\nolimits_{_{{{\rm{vs-s}}}}} $的混合样本集
    63 ${P_{{{\rm{num}}}}}$ 种群中粒子数量
    64 ${N_{{{\rm{iter}}}}}$ 种群迭代次数
    65 ${{\boldsymbol{A}}}$ 种群的外部档案, 保存非支配解.
    下载: 导出CSV

    表  2  基准数据集划分

    Table  2  Benchmark data set partitioning

    数据集 特征数 训练集 验证集 测试集 实验数据编号
    数量 $\eta $ 数量 $\eta $ 数量 $\eta $
    混凝土抗压强度 8 20 0.3327 20 0.3598 100 0.1255 A1
    40 0.2444 40 0.2628 A2
    60 0.1853 60 0.2070 A3
    超导 81 20 0.3351 20 0.3388 100 0.1538 B1
    40 0.2309 40 0.2423 B2
    60 0.1949 60 0.1966 B3
    下载: 导出CSV

    表  3  基准数据基于多目标PSO混合优化的VSG参数设定

    Table  3  Parameter setting of VSG based on hybrid optimization with multi-objective PSO for benchmark data

    数据 ${P_{{{\rm{num}}}}}$ ${N_{{{\rm{iter}}}}}$ ${N_{{{\rm{refresh}}}}}$ $K$ ${z_{{{\rm{MTD}}}}}$ $z_{{{\rm{RF}}}}^{{\rm{1}}}$ $z_{{{\rm{RF}}}}^{{\rm{2}}}$ ${z_{{{\rm{RWNN}}}}}$
    A 30 30 3 30 (0, 1) (1, 6) (2, 10) (3, 20)
    B 30 30 3 50 (0, 1) (1, 30) (2, 10) (3, 20)
    下载: 导出CSV

    表  4  基准数据基于多目标PSO混合优化获得的最优虚拟样本

    Table  4  Optimal virtual samples obtained based on multi-objective PSO hybrid optimization for benchmark data

    数据集 ${\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{vs}}}}} }$ ${y_{{{\rm{vs}}}}}$
    A1 396.5 117.4 0 176.4 11.42 876.7 796.9 60.23 58.83
    200.5 16.35 115.8 161.6 8.27 1071.7 809.9 17.23 29.23
    240.9 0 100.3 183.5 5.87 977.3 852.4 14 18.25
    272.4 56.58 0 199 0 965.0 786.9 37.38 12.62
    347.4 0 0 190.8 0 1116.4 718.2 15.08 3.42
    B1 5.69 95.64 60.78 69.89 36.85 1.481 1.410 182.2 26.79
    4.08 77.39 51.82 60.19 35.09 1.218 1.269 121.4 95.32
    4 76.44 50.35 59.37 34.71 1.200 1.291 121.3 80.12
    4.46 82.72 56.99 64.52 36.03 1.298 1.090 131.2 51.89
    3.54 83.97 60.06 66.37 43.11 1.066 0.974 99.9 6.38
    下载: 导出CSV

    表  5  基准数据原始样本输入输出范围

    Table  5  Input/output range of original samples for benchmark data

    数据集 输入 输出
    A1 最小值 102 0 0 121.75 0 801 594 1 2.33
    最大值 540 359.4 200.1 247 32.2 1145 992.6 365 82.60
    B1 最小值 1 6.9 6.4 5.3 2.0 0 0 0 0.00
    最大值 9 209.0 209.0 209.0 209.0 1.98 1.96 208.0 185
    下载: 导出CSV

    表  6  基准数据基于多目标PSO混合优化的全局最优解的统计结果

    Table  6  Statistical results of global optimal solution based on hybrid optimization with multi-objective PSO for benchmark data

    数据集 超参数 虚拟样本数量 验证集 测试集 混合样本$\eta $
    ${\gamma _{{{\rm{extend}}}}}$ ${L_F}$ ${\theta _{{{\rm{leaf}}}}}$ $I$ 平均$RMSE$ 平均$\rho $ 平均$RMSE$ 平均$\rho $
    A1 0.6033 3 9 18 82 10.36 0.026 11.59 0.012 0.2354
    A2 0.6245 6 5 19 128 10.03 0.012 10.73 0.003 0.2099
    A3 0.6528 6 9 20 150 10.40 0.006 10.28 0.002 0.2002
    B1 0.3951 5 5 16 20 16.44 0.300 19.07 0.169 0.2407
    B2 0.4892 8 6 14 69 20.14 0.019 17.86 0.051 0.2118
    B3 0.6775 19 6 15 70 19.57 0.000 18.05 0.023 0.2076
    下载: 导出CSV

    表  7  基准数据不同VSG方法的对比统计结果

    Table  7  Comparative statistical results of different VSG methods for benchmark data

    数据集 方法 虚拟样本数量 混合样本$\eta $ 测试$RMSE$ 测试$\rho $
    均值 方差 最优 均值×${10^{ - 3}}$ 方差×${10^{ - 4}}$ 最优×${10^{ - 3}}$
    A1 N-VSG[29] 219 0.2770 16.47 8.785 14.11 4.09 15.44 4.62
    M-VSG[31] 238 0.3018 17.08 8.575 13.65 2.26 19.73 4.55
    PSO-VSG[34] 55 0.4235 16.35 3.822 12.75 3.76 30.2 5.88
    MP-VSG[35] 165 0.2641 14.03 4.525 12.93 6.04 9.93 7.19
    $\bf{MoHo-VSG}$ ${\bf{82}}$ ${\bf{0.2354}}$ ${\bf{11.59}}$ ${\bf{0.107}}$ ${\bf{9.67}}$ ${\bf{12.46}}$ ${\bf{1.34}}$ ${\bf{14.72}}$
    B1 N-VSG[29] 176 0.2945 24.38 10.541 21.96 13.87 17.96 14.25
    M-VSG[31] 281 0.3100 25.33 12.786 20.12 12.63 56.11 14.12
    PSO-VSG[34] 36 0.3317 26.11 17.71 20.38 1.69 71.20 8.23
    MP-VSG[35] 134 0.2513 20.84 3.452 19.47 17.43 4.37 18.89
    $\bf{MoHo-VSG}$ ${\bf{20}}$ ${\bf{0.2076}}$ ${\bf{18.05}}$ ${\bf{0.062}}$ ${\bf{17.84}}$ ${\bf{169.26}}$ ${\bf{1.57}}$ ${\bf{178.69}}$
    下载: 导出CSV

    表  8  DXN数据基于多目标PSO混合优化的VSG算法参数设定-数据C

    Table  8  Parameter setting of VSG algorithm based on multi-objective PSO hybrid optimization for DXN data-Data C

    参数 ${P_{{{\rm{num}}}}}$ ${N_{{{\rm{iter}}}}}$ ${N_{{{\rm{refresh}}}}}$ $K$ ${z_{{{\rm{MTD}}}}}$ $z_{{{\rm{RF}}}}^{{\rm{1}}}$ $z_{{{\rm{RF}}}}^{{\rm{2}}}$ ${z_{{{\rm{RWNN}}}}}$
    数据 30 30 3 50 (0, 1) (1, 35) (2, 10) (3, 20)
    下载: 导出CSV

    表  9  DXN数据基于多目标PSO混合优化获得的最优虚拟样本-数据C

    Table  9  Optimal virtual samples obtained based on multi-objective PSO hybrid optimization for DXN data-Data C

    ${\mathop {{\boldsymbol{X}}}\nolimits_{_{{{\rm{vs}}}}} }$ ${y_{{{\rm{vs}}}}}$
    4.366 1.54 68.78 27.31 241.4 3.96 334.7 0.0289
    4.206 0 68.94 28.15 222.5 3.77 306.8 0.0458
    4.449 7.69 72.48 30.23 222.8 3.98 315.8 0.0685
    4.432 10 71.83 30 225.9 3.99 319.5 0.0163
    4.461 17.69 74.65 30.77 228.5 3.99 321.8 0.0029
    下载: 导出CSV

    表  10  DXN数据面向VSG的多目标PSO混合优化全局最优解-数据C

    Table  10  DXN data for VSG-oriented multi-objective PSO hybrid optimization global optimal solution-Data C

    性能指标 最优解
    超参数${\gamma _{{{\rm{extend}}}}}$ 0.1206
    超参数${L_F}$ 2
    超参数${\theta _{{{\rm{leaf}}}}}$ 5
    超参数$I$ 15
    虚拟样本数量 40
    验证集的平均$RMSE$ 0.0231
    验证集的平均$\rho $ 4.41×${10^{ - 5}}$
    测试集的平均$RMSE$ 0.0238
    测试集的平均$\rho $ 3.18×${10^{ - 5}}$
    小样本建模的验证$RMSE$ 0.0259
    小样本建模的测试$RMSE$ 0.0251
    下载: 导出CSV

    表  11  针对DXN数据的不同VSG方法的对比统计结果-数据C

    Table  11  Comparative statistical results of different VSG methods based on DXN dataset-Data C

    方法 虚拟样本数量 测试$RMSE$ 测试$\rho $
    均值 方差×${10^{ - 4}}$ 最优 均值×${10^{ - 5}}$ 方差 最优×${10^{ - 5}}$
    N-VSG[29] 129 0.0406 0.695 0.0262 0.19 1.94×${10^{ - 5}}$ 0.36
    M-VSG[31] 116 0.0403 1.331 0.0231 0.26 8.83×${10^{ - 5}}$ 0.53
    PSO-VSG[34] 27 0.0328 0.519 0.0245 0.56 8.44×${10^{ - 5}}$ 1.02
    MP-VSG[35] 68 0.0377 1.208 ${\bf{0.0218}}$ 1.04 5.16×${10^{ - 7}}$ 1.78
    $\bf{MoHo-VSG}$ ${\bf{40}}$ ${\bf{0.0231}}$ ${\bf{0.691}}$ 0.0220 ${\bf{3.18}}$ 4.47×${10^{ - 9}}$ ${\bf{3.45}}$
    下载: 导出CSV
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  • 收稿日期:  2021-11-18
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