Finding all real roots of 3×3 nonlinear algebraic systems using neural networks

Goulianas, Konstantinos; Margaris, Athanasios; Adamopoulos, Miltiades

doi:10.1016/j.amc.2012.10.049

Cited by 6 publications

(20 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In this way, the network was capable of estimating all the roots of the system, one root per training, according to the set of initial conditions. This network has been tested successfully in solving 2 2 [36] as well as 3 3 nonlinear systems [37]. Continuing this line of research, the present article generalizes the solvers presented in [36] and [37] for solving 2 2 [36] as well as 3 3 nonlinear systems [37] for the general case of n n nonlinear systems.…”

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 67%

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 96%

“…Continuing this line of research, the present article generalizes the solvers presented in and for solving 2 × 2 as well as 3 × 3 nonlinear systems for the general case of n × n nonlinear systems. This generalization is not restricted only to the dimension of the neural solver, but it is extended to the form of the system because it has been enhanced with some new features that do not appear in and . The most important feature is that the activation functions of the third layer neurons can be any continuous and differentiable function, a feature that adds another degree of nonlinearity and allows the network to work correctly and with success, even in cases where the activation function of the output neurons is a simple linear function such as the identity functions.…”

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 96%

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 99%

See 3 more Smart Citations

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

Goulianas

Margaris

Refanidis

et al. 2015

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper presents an MLP‐type neural network with some fixed connections and a backpropagation‐type training algorithm that identifies the full set of solutions of a complete system of nonlinear algebraic equations with n equations and n unknowns. The proposed structure is based on a backpropagation‐type algorithm with bias units in output neurons layer. Its novelty and innovation with respect to similar structures is the use of the hyperbolic tangent output function associated with an interesting feature, the use of adaptive learning rate for the neurons of the second hidden layer, a feature that adds a high degree of flexibility and parameter tuning during the network training stage. The paper presents the theoretical aspects for this approach as well as a set of experimental results that justify the necessity of such an architecture and evaluate its performance. Copyright © 2015 John Wiley & Sons, Ltd.

show abstract

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 67%

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 96%

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 96%

Section: The Structure Of the Proposed Neural Nonlinear System Solvermentioning

confidence: 99%

See 2 more Smart Citations

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

Goulianas

Margaris

Refanidis

et al. 2015

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Theorem 12. Assume that conditions ( 1 ), ( 2 ), and ( 3 ) hold; then problem (25) or (26) has at least a positive solution ∈ ∩ (Ω \ Ω ). Such solution satisfies the condition ≤ | | ≤ .…”

Section: More General Resultsmentioning

confidence: 99%

Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Zhang

Feng

2016

Mathematical Problems in Engineering

View full text Add to dashboard Cite

Based on Guo-Krasnoselskii’s fixed point theorem, the existence of positive solutions for a class of nonlinear algebraic systems of the formx=GFxis studied firstly, whereGis a positiven×nsquare matrix,x=col⁡(x1,x2,…,xn), andF(x)=col⁡(f(x1),f(x2),…,f(xn)), where,F(x)is not required to be satisfied sublinear or superlinear at zero point and infinite point. In addition, a new cone is constructed inRn. Secondly, the obtained results can be extended to some more general nonlinear algebraic systems, where the coefficient matrixGand the nonlinear term are depended on the variablex. Corresponding examples are given to illustrate these results.

show abstract

Solving polynomial systems using a fast adaptive back propagation-type neural network algorithm

Goulianas¹,

Margaris

Refanidis

et al. 2017

Eur. J. Appl. Math

View full text Add to dashboard Cite

This paper proposes a neural network architecture for solving systems of non-linear equations. A back propagation algorithm is applied to solve the problem, using an adaptive learning rate procedure, based on the minimization of the mean squared error function defined by the system, as well as the network activation function, which can be linear or non-linear. The results obtained are compared with some of the standard global optimization techniques that are used for solving non-linear equations systems. The method was tested with some well-known and difficult applications (such as Gauss–Legendre 2-point formula for numerical integration, chemical equilibrium application, kinematic application, neuropsychology application, combustion application and interval arithmetic benchmark) in order to evaluate the performance of the new approach. Empirical results reveal that the proposed method is characterized by fast convergence and is able to deal with high-dimensional equations systems.

show abstract

Finding all real roots of 3×3 nonlinear algebraic systems using neural networks

Cited by 6 publications

References 13 publications

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

An adaptive learning rate backpropagation‐type neural network for solving n × n systems on nonlinear algebraic equations

Existence of Positive Solutions for a Class of Nonlinear Algebraic Systems

Solving polynomial systems using a fast adaptive back propagation-type neural network algorithm

Contact Info

Product

Resources

About