2024年5月23日发(作者：)

摘 要

复值神经网络是一类用复数变量解决复杂问题的网络。梯度下降法是训练复值神经

网络的流行算法之一。目前，建立的传统网络模型大多数是整数阶模型。与经典的整数

阶模型相比，建立在分数阶微积分上的模型在记忆储存和遗传特性上都具有显著的优势。

源于分数阶微分系统特性和复数的几何意义，分数阶复值神经网络具有比整数阶复值情

形更为优越的记忆特性。本文基于分离复值神经网络，利用分数阶导数来训练分离复值

神经网络（Split-complex neural networks, SCNNs）。根据误差函数关于权值的梯度的不

同定义，提出了两种权值更新方法。借助微分中值定理和不等式分析技巧，并且在常学

习率下，证明了分数阶复值神经网络学习算法在训练迭代中的误差函数是单调递减的，

并且误差函数关于权值的梯度趋于零。此外，数值模拟已经有效地验证了其性能，同时

也说明了理论结果。本文主要工作如下：

1. 提出了分数阶复值神经网络算法。通过使用分数阶最陡下降法 (FSDM)，描述

了基于Faà di Bruno’s formula的分数阶复值神经网络(FCBPNNs)的实现。

2. 提出一种基于Caputo定义的分数阶复值神经网络学习算法。在适合的激活函数

和学习率的条件下，结合分离复值BP算法和Caputo微分定义形式，利用分数

阶最速下降学习算法训练网络权值，得到了该算法误差函数的单调递减性质。

3. 严格证明基于Caputo定义的复值神经网络学习算法的收敛性，这样就从理论的

角度保证了该算法的收敛行为。

4. 基于Caputo定义的FCBPNNs，选择经典的二维奇偶问题进行数值实验，对不

同分数阶次与整数阶的复值BP神经网络的误差函数曲线图进行对比，同时也验

证了理论结果。

关键词：复值神经网络；分数阶导数；单调性；收敛性

i

Algorithm Design and Analysis for fractional order complex-valued

neural networks

Yang Guoling (college of science)

Directed by Associate Prof. Wang Jian

Abstract

The complex-valued neural networks are the class of networks that solve complex

problems by using complex-valued variables. The gradient descent method is one of the

popular algorithms to train complex-valued neural networks. At present, most of the

established networks are integer-order models. Compared with classical integer-order models,

the built models in terms of fractional calculus possess significant advantages on both

memory storage and hereditary characteristics. Derived from the properties of fractional

differential systems and geometric meaning of complex number, fractional-valued complex

neural network has better memory property than integer-valued complex neural network. This

paper is developed on the model of split-complex neural network (SCNNs). The fractional

derivative is used to train split-complex neural networks. According to the definition of the

gradient of the error function with respect to weight, two methods of updating the weight are

proposed. The monotonicity of the proposed algorithms has been obtained based on the

calculus mean value theorem and inequality analysis technique. Furthermore, the norms of the

gradient of the error function with respect to weights are proven to be close to zero as the

iteration approaches infinite. This property guarantees the deterministic convergence of the

proposed algorithm from a mathematical point of view. In addition, numerical simulation has

effectively verified its competitive performance and also illustrated the theoretical results.

The main contributions of this work are as follows:

1. A fractional-order complex neural network algorithm is proposed. By using FSDM, a

fractional-order complex valued neural network (FCBPNNs) based on Faà di

Bruno’s formula is described .

ii