Fractional-Order Non-causal BP Neural Networks Model

doi:10.3778/j.issn.1002-8331.2012-0072

Abstract

Abstract:

Based on the definition of the first-order causal and the first-order anti-causal derivative, the anti-causal Caputo derivative is deduced. Then a non-causal Caputo derivative can be given. By applying them to the back propagation process of BP neural network to deal with the weight, the causal, anti-causal and non-causal BP neural network model are generated. For the convenience of comparison, these models are processed for the Boston Housing dataset and the MNIST dataset separately. The results of the simulation show that the integer-order, non-causal model has the best results among the integer-order models; Fractional-order causal, anti-causal and non-causal models are compared with their corresponding integer-order models separately, then the results of the fractional- order models are better. The accuracy of the non-causal is the highest among the fractional-order models. In general, Caputo causal, anti-causal and non-causal calculus all have positive influence on BP neural network, especially the fractional-order non-causal calculus has the best effect.

Key words: causal, anti-causal, non-causal, Caputo derivative, BP neural network

摘要：

由一阶因果、反因果微分的定义推导出Caputo分数阶因果、反因果微积分，并在此基础上定义Caputo分数阶非因果微积分。将它们分别应用于BP神经网络的反向传播过程中对权值进行处理，产生了Caputo分数阶因果、反因果和非因果BP神经网络模型。为了方便对比，将这些模型分别对波士顿房屋数据集和MNIST数据集进行处理。模拟结果表明：在整数阶因果、反因果和非因果的模型之间，整数阶非因果模型的结果最好；分数阶因果、反因果和非因果模型分别与其相应的整数阶模型进行比较，得出分数阶模型得到的准确率比整数阶的高；在分数阶因果、反因果和非因果的模型之间，非因果的准确性最高。总的来说，Caputo分数阶因果、反因果和非因果微积分都对传统BP神经网络有优化作用，尤其是分数阶非因果微积分的优化效果最好。

关键词: 因果, 反因果, 非因果, Caputo分数阶微积分, BP神经网络

HUANG Jingjing, WANG Jianhong. Fractional-Order Non-causal BP Neural Networks Model[J]. Computer Engineering and Applications, 2021, 57(23): 91-97.

黄晶晶，王建宏. 分数阶非因果BP神经网络模型[J]. 计算机工程与应用, 2021, 57(23): 91-97.

References

[1] 刘承伟，赵洪凯，严豪，等.基于分数阶GM（1，1）与BP神经网络的电力负荷预测[J].数学的实践与认识，2018，48（23）：145-151.
LIU C W，ZHAO H K，YAN H，et al.Power load forecasting based on fractional GM（1，1） and BP neural network[J].Mathematics in Practice and Theory，2018，48（23）：145-151.
[2] 易云清，吕乐群，卢圆圆，等.基于神经网络的复杂信号样式调制识别技术[J].电子信息对抗技术，2020，35（6）：16-21.
YI Y Q，LV L Q，LU Y Y，et al.The modulation recognition technology of complex signal pattern based on neural network[J].Electronic Warfare Technology，2020，35（6）：16-21.
[3] 何明慧，徐怡，王冉，等.改进的粒子群算法优化神经网络及应用[J].计算机工程与应用，2018，54（19）：107-113.
HE M H，XU Y，WANG R，et al.Improved particle swarm optimization neural network and its application[J].Computer Engineering and Applications，2018，54（19）：107-113.
[4] 罗宇卓.基于FPS0优化的BP神经网络算法研究与应用[D].银川：宁夏大学，2018.
LUO Y Z.Research on BP neural network algorithm and application based on FPSO optimization[D].Yinchuan：Ningxia University，2018.
[5] WANG J，WEN Y，GOU Y，et al.Fractional-order gradient descent learning of BP neural networks with Caputo derivative[J].Neural Networks，2017，89（89）：19-30.
[6] 温艳青.分数阶神经网络学习算法的设计与理论分析[D].青岛：中国石油大学，2017.
WEN Y Q.Algorithm design and convergence analysis for fractional-order neural network[D].Qingdao：China University of Petroleum，2017.
[7] 付华，王福娇，陈子春.基于分数阶神经网络的瓦斯涌出量预测[J].传感器与微系统，2013，32（5）：31-34.
FU H，WANG F J，CHEN Z C.Prediction of gas emission based on fractional order neural network[J].Transducer and Microsystem Technologies，2013，32（5）：31-34.
[8] DAS S.Fractional calculus[M].Berlin：Springer-Verlag，2011.
[9] SAMKO S，KILBAS A，MARICHEV O.Fractional integrals and derivatives：theory and applications[M].New York： Gordon and Breach Science Publishers，1993.
[10] 吴媛媛，潘祥，姜太平，等.基于非因果信号处理的分数阶次低通滤波器[J].计算机应用与软件，2017，34（9）：192-198.
WU Y　Ｙ，PAN　X，JIANG　Ｔ　Ｐ，et al.Non-causal signal processing based fractional low-pass Filter[J].Computer Applications and Software，2017，34（9）：192-198.
[11] 刘双双.基于非因果信号处理的分数阶次图像去噪算法研究[D].马鞍山：安徽工业大学，2017.
LIU S S.Non-causal signal processing based fractional image denoising algorithm[D].Ma’anshan：Anhui University of Technology，2017.
[12] TSAI M，LIN M，YAU H.Development of command-based iterative learning control algorithm with consi-deration of friction，disturbance，and noise effects[J].IEEE Transactions on Control Systems Technology，2006，14（3）：511-518.
[13] 朱洪俊.心电信号零相位数字滤波[J].北京生物医学工程，2003（4）：260-262.
ZHU H J.ECG signal processing with zero-phase digital filtering[J].Beijing Biomedical Engineering，2003（4）：260-262.
[14] WANG J，YE Y，GAO X.Fractional 90° phase-shift filtering based on the double-sided Grünwald-Letnikov differintegrator[J].IET Signal Processing，2015，9（4）：328-334.
[15] ANDREI V.Neural network for low-memory IoT devices and MNIST image recognition using kernels based on logistic map[J].Electronics，2020，9（9）：1432.