西 南 交 通 大 学 学 报
第 55 卷 第 1 期
2020 年 2 月
JOURNAL OF SOUTHWEST JIAOTONG UNIVERSITY
Vol. 55 No. 1
Feb. 2020
ISSN: 0258-2724 DOI：10.35741/issn.0258-2724.55.1.41
Research articleMathematics
S
OLVING A
S
YSTEM OF
S
INGULAR
P
ERTURBATION
P
ROBLEMS VIA
N
EURAL
N
ETWORKS
通过神经网络解决奇异摄动问题系统
Rana Taleb Shwayaa a, Manar Joundy Hazar b, Mustafa Radif c, Jmal Aldeen Dahi d
a Department of Mathematics, College of Education, University of Al-Qadisiyah P.O. Box 88, Al Diwaniyah, AL-Qadisiyah, Iraq, [email protected]
b
Computer Center, University of Al-Qadisiyah
P.O. Box 88, Al Diwaniyah, AL-Qadisiyah, Iraq, [email protected]
c College of Computer Science and Information Technology, University of Al-Qadisiyah P.O. Box 88, Al Diwaniyah, AL-Qadisiyah, Iraq, [email protected]
d
College of Dentistry, University of Al-Qadisiyah
P.O. Box 88, Al Diwaniyah, AL-Qadisiyah, Iraq, [email protected]
Abstract
In this work, we will introduce a new procedure to solve a system of singular perturbation problems (SSPPs) via artificial neural networks. The neural networks use the code of back propagation with altered training algorithms such as quasi-Newton, Levenberg-Marquardt, and Bayesian regularization. In our research, we provide examples of two different types of systems, showing the accuracy, speed, resolution, and convergence of the new technology, the effectiveness of using the network techniques for solving this type of equations. The convergence properties of the technique and accuracy of the interpolation technique are considered.
Keywords: Singular Perturbation Problem, Neural Network, Levnberg-Marquardt
摘要 在这项工作中，我们将介绍一种通过人工神经网络解决奇异摄动问题（SSPPs）系统的新过程。 神经网络将反向传播代码与经过更改的训练算法（如准牛顿，莱文贝格-马夸特和贝叶斯正则化）结 合使用。 在我们的研究中，我们提供了两种不同类型的系统的示例，这些示例显示了新技术的准确 性，速度，分辨率和收敛性，以及使用网络技术求解此类方程式的有效性。考虑了该技术的收敛性和 内插技术的准确性。 关键词: 奇摄动问题，神经网络，莱文贝格-马夸特
I.
I
NTRODUCTIONThe idea for this paper came from previous research by Tawfiq and Al-Abrahemee, which attempted to solve one singular perturbation problem equation [1]. This work has been developed to be a system of particular perturbation problems. Several problems ascending from the actual design cannot be resolved fully by analytical solution. One of the best imperative difficulties ascending in some other sciences is SPPs. The SPPs are measured in differential equations wherever the main derivatives are increased in lesser constraints, which are a central subcategory of stiff problems. These difficulties need to be regularly resolved arithmetically; conversely, for they are stiff and the answer is contingent on the lesser constraint, (naturally piece boundary layers), the numerical behavior of these difficult offerings definite main computational complications. Some structures have been advanced for the arithmetical answer of SPPs [2], [3], [4], [5], [6]. However, there is a certain event of s in which the answers are efficiently variable and ε-independent (parameter
ε), while the problems become very stiff as the
small parameter, ε, tends to zero [7], [8], [9]. The aim of this study is to solve the system of system of singular perturbation problems with a new technology, parallel processing technology via neural networks, and the neural network is Levenberg-Marquardt algorithm (LM).
II.
S
INGULARP
ERTURBATIONP
ROBLEMS[10]
Perturbation theory is an assembly of ways for finding estimated answers to difficulties, including a small parameter . These methods are very powerful, thus, occasionally, it is essentially suitable to present a parameter ε momentarily into a problem, taking no small parameter, and then lastly to set = 1 to improve the new problem. The method of perturbation theory is to fester a rough problem into a (unlimited) number of comparatively informal ones. Perturbation theory is most expedient when the first few steps reveal the imperative structures of the answer and the lasting ones provide small improvements. We order perturbation answers into two varieties. A simple article of consistent perturbation problems is that the exact answer for small but nonzero ε smoothly approaches the composed solution as → 0. We define a singular perturbation problem as one whose answer for = 0 is essentially altered in attraction from the “neighboring” answers obtained in the bound → 0.
III.
L
EVENBERG-M
ARQUARDTA
LGORITHM(LM)
[11]
The performance indicator to be improved is defined as
(1) where consists of all weights of the network, is the chosen importance of the yield and the pattern, is the definite importance of the yield and the form, is the numeral of the weights, is the number of the form, and is the numeral of the network yield.
Equation (1) could be
(2) where
where is the increasing error vector (for all forms) from equation (2), the Jacobian matrix is defined as
= (3)
Weight can be calculated through the following equation:
(4) Such that I is identified as the unit matrix, is the learning consideration, and J is Jacobian of m yield error. The consideration is robotically attuned at all iterations in the command to protect the conjunction; the algorithm needs
computation
the Jacobian matrix at each iteration stage and the transposition of the square matrix, the length of which is * .IV.
I
LLUSTRATION OF THEM
ETHODIn this section, we will show how our method can be used to find the estimated answer to the
overall formula—a first order of a system of ordinary singular perturbation problems (SSPPs). (5)
Such that perturbation parameter with , (i = 1, . . . ,K), we consider one neural network for each trial solution (i =1, . . . ,K), which is written as follows:
(6)
We minimize the following error quantity:
(7)
V.
N
UMERICALE
XAMPLESIn this section of the paper, we will take two examples of two different cases of systems of perturbation equations of diverse values of the
perturbed parameter and then compare the approximate solutions of the offered method with analytical solutions to show the accuracy of the method and the speed of convergence.
A. Example 1 [7]
Let the SPPs
B. Example 2
Let the SPPs [8]
The exact solution of this problem is as follows:
