Neural network method: delay and system of delay differential equations

Panghal, Shagun; Kumar, Manoj

doi:10.1007/s00366-021-01373-z

Cited by 17 publications

(6 citation statements)

References 25 publications

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“…Example 7.1 25 Consider the pantograph equation with variable coefficients and multiple delays where, g(t) = 1 8 e −t (12sin(t)…”

Section: Comparative Analysismentioning

confidence: 99%

“…The maximum relative error of a simple feed-forward neural network(FNN) method in 25 is 4.05 × 10 −10 and the maximum relative error of the proposed FLNN-based ONN method is 3.40 × 10 −11 . This comparison shows that the ONN method can obtain a better accuracy solution than simple FNN.…”

Section: Comparative Analysismentioning

confidence: 99%

“…The maximum relative error of a simple feed-forward neural network(FNN) method in 25 for z 1 (t) and z 2 (t) with twelve training points are 1.93 × 10 −9 and 2.42 × 10 −9 respectively and the maximum relative error of the proposed FLNN-based ONN method for z 1 (t) and z 2 (t) with twelve training points are 1.60 × 10 −9 and 5.11 × 10 −10 respectively. This comparison shows that the ONN method can obtain a better accuracy solution than simple FNN.…”

Section: Comparative Analysismentioning

confidence: 99%

“…The maximum relative error of a simple feed-forward neural network(FNN) method in 25 for z 1 (t) , z 2 (t) and z 3 (t) with ten training points are 8.78 × 10 −8 , 1.42 × 10 −8 and 1.93 × 10 −7 respectively and the maximum relative error of the proposed FLNN-based ONN method for z 1 (t) , z 2 (t) and z 3 (t) with ten training points are…”

Section: Comparative Analysismentioning

confidence: 99%

See 3 more Smart Citations

Numerical solution of neutral delay differential equations using orthogonal neural network

Vinodbhai

Dubey

2023

Sci Rep

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In this paper, an efficient orthogonal neural network (ONN) approach is introduced to solve the higher-order neutral delay differential equations (NDDEs) with variable coefficients and multiple delays. The method is implemented by replacing the hidden layer of the feed-forward neural network with the orthogonal polynomial-based functional expansion block, and the corresponding weights of the network are obtained using an extreme learning machine(ELM) approach. Starting with simple delay differential equations (DDEs), an interest has been shown in solving NDDEs and system of NDDEs. Interest is given to consistency and convergence analysis, and it is seen that the method can produce a uniform closed-form solution with an error of order $$2^{-n}$$ 2 - n , where n is the number of neurons. The developed neural network method is validated over various types of example problems(DDEs, NDDEs, and system of NDDEs) with four different types of special orthogonal polynomials.

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“…Example 7.1 25 Consider the pantograph equation with variable coefficients and multiple delays where, g(t) = 1 8 e −t (12sin(t)…”

Section: Comparative Analysismentioning

confidence: 99%

Section: Comparative Analysismentioning

confidence: 99%

Section: Comparative Analysismentioning

confidence: 99%

Section: Comparative Analysismentioning

confidence: 99%

See 2 more Smart Citations

Numerical solution of neutral delay differential equations using orthogonal neural network

Vinodbhai

Dubey

2023

Sci Rep

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show abstract

“…In this order, the feed‐forward neural network has been applied to solve the delay and system of delay ordinary differential equations [34]. In [35], a deep neural network was trained to solve the coupled system of Emden‐Fowler equations, and that network achieved an accuracy of up to 10 −6 .…”

Section: Introductionmentioning

confidence: 99%

Multilayered neural network for power series‐based approximation of fractional delay differential equations

Kumar,

Kumar

et al. 2024

Math Methods in App Sciences

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This paper trains a multilayered neural network (MLNN) for solving fractional delay differential equations (FDDEs), including nonlinear and singular types. The proposed methodology involves replacing the unknown functions in the equations with a truncated power series expansion. Subsequently, a collection of algebraic equations is solved utilizing an iterative minimization technique that leverages the capabilities of the MLNN architecture. The outcomes demonstrate that the MLNN architecture provides the required accuracy and strong stability compared to several numerical methods.

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Transfer physics informed neural network: a new framework for distributed physics informed neural networks via parameter sharing

Manikkan

Srinivasan

2022

Engineering with Computers

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Neural network method: delay and system of delay differential equations

Cited by 17 publications

References 25 publications

Numerical solution of neutral delay differential equations using orthogonal neural network

Numerical solution of neutral delay differential equations using orthogonal neural network

Multilayered neural network for power series‐based approximation of fractional delay differential equations

Transfer physics informed neural network: a new framework for distributed physics informed neural networks via parameter sharing

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