Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation

Raja, Muhammad Asif Zahoor; Samar, Raza; Rashidi, Majid

doi:10.1007/s00521-014-1641-x

Cited by 26 publications

(6 citation statements)

References 33 publications

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“…The non-conventional techniques based on applied soft computing methodologies have not been extensively exploited in this domain, although ANNs have been applied to solve initial and boundary value problems of ordinary and partial differential equations effectively [15][16][17]. As a reference, reliable solution for nonlinear Van der Pol oscillators with stiff as well as non-stiff conditions [18,19], nonlinear singular system based on Lane-Emden type equations [20], nonlinear Schrodinger equations [21], fluid flow problems based on nonlinear boundary value problems (BVPs) of Jeffery-Hamel flow equations in the presence of high magnetic field [22,23], fuel ignition model of combustion theory governed by one-dimensional nonlinear BVPs of Bratu's equations [24,25], nonlinear Troesch's BVPs, a fundamental problem arises in the field of plasma physics based on confinement of a plasma column by radiation pressure and also appeared in the theory of gas porous electrodes [26,27], transformed approach for two-dimensional nonlinear Bratu's equation [28][29][30], BVPs of pantograph functional differential equations [31] and exactly satisfying the initial condition neural networks for nonlinear Painlevé I equation [32]. A survey article [33] recently published on applications of neural networks recently summarized the history, importance, recent trends and anticipated future research directions in this field.…”

Section: Introductionmentioning

confidence: 99%

Comparison of three unsupervised neural network models for first Painlevé Transcendent

Raja

Khan

Shah

et al. 2014

Neural Comput & Applic

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In this paper, a reliable soft computing framework is presented for the approximate solution of initial value problem (IVP) of first Painlevé equation using three unsupervised neural network models optimized with sequential quadratic programming (SQP). These mathematical models are constructed in the form of feed-forward architecture including log-sigmoid, radial base and tansigmoid activation functions in the hidden layers. The optimization of designed parameters for each model is performed with SQP, an efficient constraint optimization problem-solving algorithm. The designed methodology is tested on the IVP, and comparative study is carried out with standard solution based on numerical and analytical solvers. The accuracy, convergence and effectiveness of the schemes are validated on the given benchmark problem by large number of simulations and their comprehensive analysis.

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Section: Introductionmentioning

confidence: 99%

Comparison of three unsupervised neural network models for first Painlevé Transcendent

Raja

Khan

Shah

et al. 2014

Neural Comput & Applic

Self Cite

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“…However, artificial intelligence (AI) techniques have been largely used for finding the solution of initial value problems (IVPs) as well as boundary value problems (BVPs) of both linear and nonlinear type of differential equations [30][31][32][33]. Few recent applications in this domain are stochastic numerical of nonlinear Jeffery-Hamel flow study in the presence of high magnetic field [34], problems arising in electromagnetic theory [35], modelling of electrical conducting solids [36], fuel ignition type model working in combustion theory [37], magnetohydrodynamics (MHD) studies [38], fluid mechanics problems [39], drainage problem [40], plasma physics problems [41], Bratu's problems [42], Van-der-Pol oscillatory problems [43], Troesch's problems [44], nanofluidic problems [45], multiwalled carbon nanotubes studies [46], nonlinear Painleve systems [47], nonlinear pantograph systems [48] and nonlinear singular systems [49][50][51][52][53]. Furthermore, the extended form of these methods has been applied to compute the solution of linear and nonlinear well-known fractional differential equations [54,55].…”

Section: Introductionmentioning

confidence: 99%

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

Ahmad

Raja

Bilal

et al. 2016

Neural Comput & Applic

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112

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In the present study, stochastic numerical computing approach is developed by applying artificial neural networks (ANNs) to compute the solution of Lane-Emden type boundary value problems arising in thermodynamic studies of the spherical gas cloud model. ANNs are used in an unsupervised manner to construct the energy function of the system model. Strength of efficient local optimization procedures based on active-set (AS), interior-point (IP) and sequential quadratic programming (SQP) algorithms is used to optimize the energy functions. The performance of all three design methodologies ANN-AS, ANN-IP and ANN-SQP is evaluated on different nonlinear singular systems. The effectiveness of the proposed schemes in terms of accuracy and convergence is established from the results of statistical indicators.

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“…In this process, a specific connection weight corresponding to the actual output is obtained. This approach is frequently used to solve nonlinear problems (Monterola & Saloma, 2001 ; Raja et al, 2014 ) and differential equations (Parisi et al, 2003 ; Shirvany et al, 2009 ; Yadav et al, 2015 ).…”

Section: Introductionmentioning

confidence: 99%

A computational study on the optimization of transcranial temporal interfering stimulation with high‐definition electrodes using unsupervised neural networks

Bahn

Lee

Kang

2022

Human Brain Mapping

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Transcranial temporal interfering stimulation (tTIS) can focally stimulate deep parts of the brain related to specific functions using beats at two high frequencies that do not individually affect the human brain. However, the complexity and nonlinearity of the simulation limit it in terms of calculation time and optimization precision. We propose a method to quickly optimize the interfering current value of high‐definition electrodes, which can finely stimulate the deep part of the brain, using an unsupervised neural network (USNN) for tTIS. We linked a network that generates the values of electrode currents to another network, which is constructed to compute the interference exposure, for optimization by comparing the generated stimulus with the target stimulus. Further, a computational study was conducted using 16 realistic head models. We also compared tTIS with transcranial alternating current stimulation (tACS), in terms of performance and characteristics. The proposed method generated the strongest stimulation at the target, even when targeting deep areas or performing multi‐target stimulation. The high‐definition tTISl was less affected than tACS by target depth, and mis‐stimulation was reduced compared with the case of using two‐pair inferential stimulation in deep region. The optimization of the electrode currents for the target stimulus could be performed in 3 min. Using the proposed USNN for tTIS, we demonstrated that the electrode currents of tTIS can be optimized quickly and accurately. Moreover, we confirmed the possibility of precisely stimulating the deep parts of the brain via transcranial electrical stimulation.

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Application of three unsupervised neural network models to singular nonlinear BVP of transformed 2D Bratu equation

Cited by 26 publications

References 33 publications

Comparison of three unsupervised neural network models for first Painlevé Transcendent

Comparison of three unsupervised neural network models for first Painlevé Transcendent

Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model

A computational study on the optimization of transcranial temporal interfering stimulation with high‐definition electrodes using unsupervised neural networks

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