Neural algorithm for solving differential equations

Lee, Hyuk; Kang, In Seok

doi:10.1016/0021-9991(90)90007-n

Cited by 358 publications

(202 citation statements)

References 4 publications

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“…The solution of a linear system of equations is mapped onto the architecture of a Hopfield neural network. The minimization of the network's energy function provides the solution to the system of equations [2,5,6].…”

Section: Introductionmentioning

confidence: 99%

Artificial neural networks for solving ordinary and partial differential equations

Lagaris

Likas

Fotiadis

1998

IEEE Trans. Neural Netw.

1,971

1,220

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Section: Introductionmentioning

confidence: 99%

Artificial neural networks for solving ordinary and partial differential equations

Lagaris

Likas

Fotiadis

1998

IEEE Trans. Neural Netw.

1,971

1,220

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“…As such are Artificial Neural Network (AN N ) based models are used to solve ordinary differential equations with initial conditions. Lee & Kang [24] initially introduced a method to solve first order differential equation using Hopfield neural network models. Then, another approach by Meade & Fernandez [25,26] was proposed for both linear and non-linear differential equations using Splines and feed forward neural network.…”

Section: Literature Reviewmentioning

confidence: 99%

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

Khalid¹,

Sultana²,

Zaidi³

2014

IJCA

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In the current paper, a neural network method to solve sixth-order differential equations and their boundary conditions has been presented. The idea this method incorporates is to integrate knowledge about the differential equation and its boundary conditions into neural networks and the training sets. Neural networks are being used incessantly to solve all kinds of problems hailing a wide range of disciplines. Several examples are given to illustrate the efficiency and implementation of the Neural Network method.

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“…[29] discusses differences between the exact solution and approximation solutions of ODEs. [30] applies dynamics neural networks to approximate firstorder ODE. There are few works on FDE.…”

Section: Introductionmentioning

confidence: 99%

Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

Jafari¹,

Yu²,

Li³

et al. 2017

IJCIS

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The uncertain nonlinear systems can be modeled with fuzzy equations or fuzzy differential equations (FDEs) by incorporating the fuzzy set theory. The solutions of them are applied to analyze many engineering problems. However, it is very difficult to obtain solutions of FDEs.In this paper, the solutions of FDEs are approximated by two types of Bernstein neural networks. Here, the uncertainties are in the sense of Z-numbers. Initially the FDE is transformed into four ordinary differential equations (ODEs) with Hukuhara differentiability. Then neural models are constructed with the structure of ODEs. With modified back propagation method for Z-number variables, the neural networks are trained. The theory analysis and simulation results show that these new models, Bernstein neural networks, are effective to estimate the solutions of FDEs based on Z-numbers.

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Neural algorithm for solving differential equations

Cited by 358 publications

References 4 publications

Artificial neural networks for solving ordinary and partial differential equations

Artificial neural networks for solving ordinary and partial differential equations

Numerical Solution of Sixth-Order Differential Equations Arising in Astrophysics by Neural Network

Numerical Solution of Fuzzy Differential Equations with Z-numbers Using Bernstein Neural Networks

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