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Panghal S, Kumar M. A multilayer perceptron neural network approach for the solution of hyperbolic telegraph equations. NETWORK (BRISTOL, ENGLAND) 2021; 32:65-82. [PMID: 34974795 DOI: 10.1080/0954898x.2021.2015005] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/16/2019] [Revised: 09/28/2021] [Accepted: 11/30/2021] [Indexed: 06/14/2023]
Abstract
Neural networks have been extensively used for solving differential equations in the past, but they rely mostly on computationally expensive gradient-based numerical optimization procedure for solving differential equations. In this work, we are introducing a faster way to train neural networks for solving differential equations based on extreme learning machine algorithm. This algorithm is much faster as compared to traditional approaches, and it also provides highly accurate results. Reliability of the approach is tested by solving various cases of the hyperbolic telegraph equations. Solutions so obtained are compared to the results existing in the literature for analysing the accuracy of the proposed approach.
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Affiliation(s)
- Shagun Panghal
- Motilal Nehru National Institute of Technology Allahabad, Prayagraj India, 211004
| | - Manoj Kumar
- Motilal Nehru National Institute of Technology Allahabad, Prayagraj India, 211004
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2
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Panghal S, Kumar M. Approximate Analytic Solution of Burger Huxley Equation Using Feed-Forward Artificial Neural Network. Neural Process Lett 2021. [DOI: 10.1007/s11063-021-10508-8] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
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3
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Jadoon I, Ahmed A, ur Rehman A, Shoaib M, Raja MAZ. Integrated meta-heuristics finite difference method for the dynamics of nonlinear unipolar electrohydrodynamic pump flow model. Appl Soft Comput 2020. [DOI: 10.1016/j.asoc.2020.106791] [Citation(s) in RCA: 29] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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4
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Tan LS, Zainuddin Z, Ong P. Wavelet neural networks based solutions for elliptic partial differential equations with improved butterfly optimization algorithm training. Appl Soft Comput 2020. [DOI: 10.1016/j.asoc.2020.106518] [Citation(s) in RCA: 18] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Mehmood A, Chaudhary NI, Zameer A, Raja MAZ. Backtracking search optimization heuristics for nonlinear Hammerstein controlled auto regressive auto regressive systems. ISA TRANSACTIONS 2019; 91:99-113. [PMID: 30770155 DOI: 10.1016/j.isatra.2019.01.042] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/02/2018] [Revised: 12/13/2018] [Accepted: 01/31/2019] [Indexed: 06/09/2023]
Abstract
In this work, novel application of evolutionary computational heuristics is presented for parameter identification problem of nonlinear Hammerstein controlled auto regressive auto regressive (NHCARAR) systems through global search competency of backtracking search algorithm (BSA), differential evolution (DE) and genetic algorithms (GAs). The mean squared error metric is used for the fitness function of NHCARAR system based on difference between actual and approximated design variables. Optimization of the cost function is conducted with BSA for NHCARAR model by varying degrees of freedom and noise variances. To verify and validate the worth of the presented scheme, comparative studies are carried out with its counterparts DE and GAs through statistical observations by means of weight deviation factor, root of mean squared error, and Thiel's inequality coefficient as well as complexity measures.
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Affiliation(s)
- Ammara Mehmood
- Department of Electrical Engineering, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan.
| | | | - Aneela Zameer
- Department of Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad, Pakistan.
| | - Muhammad Asif Zahoor Raja
- Department of Electrical and Computer Engineering, COMSATS University Islamabad, Attock Campus, Attock, Pakistan.
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6
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Integrated intelligent computing paradigm for the dynamics of micropolar fluid flow with heat transfer in a permeable walled channel. Appl Soft Comput 2019. [DOI: 10.1016/j.asoc.2019.03.026] [Citation(s) in RCA: 51] [Impact Index Per Article: 10.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/19/2022]
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Raja MAZ, Asma K, Aslam MS. Bio-inspired computational heuristics to study models of HIV infection of CD4+ T-cell. INT J BIOMATH 2018. [DOI: 10.1142/s1793524518500195] [Citation(s) in RCA: 31] [Impact Index Per Article: 5.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
Abstract
In this work, biologically-inspired computing framework is developed for HIV infection of CD4[Formula: see text] T-cell model using feed-forward artificial neural networks (ANNs), genetic algorithms (GAs), sequential quadratic programming (SQP) and hybrid approach based on GA-SQP. The mathematical model for HIV infection of CD4[Formula: see text] T-cells is represented with the help of initial value problems (IVPs) based on the system of ordinary differential equations (ODEs). The ANN model for the system is constructed by exploiting its strength of universal approximation. An objective function is developed for the system through unsupervised error using ANNs in the mean square sense. Training with weights of ANNs is carried out with GAs for effective global search supported with SQP for efficient local search. The proposed scheme is evaluated on a number of scenarios for the HIV infection model by taking the different levels for infected cells, natural substitution rates of uninfected cells, and virus particles. Comparisons of the approximate solutions are made with results of Adams numerical solver to establish the correctness of the proposed scheme. Accuracy and convergence of the approach are validated through the results of statistical analysis based on the sufficient large number of independent runs.
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Affiliation(s)
- Muhammad Asif Zahoor Raja
- Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock Campus, Attock, Pakistan
| | - Kiran Asma
- Department of Computer Sciences, COMSATS Institute of Information Technology, Attock Campus, Attock, Pakistan
| | - Muhammad Saeed Aslam
- Pakistan Institute of Engineering and Applied Sciences, Nilore Islamabad, Pakistan
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Chen D, Zhang Y, Li S. Zeroing neural-dynamics approach and its robust and rapid solution for parallel robot manipulators against superposition of multiple disturbances. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.09.032] [Citation(s) in RCA: 46] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Raja MAZ, Abbas S, Syam MI, Wazwaz AM. Design of neuro-evolutionary model for solving nonlinear singularly perturbed boundary value problems. Appl Soft Comput 2018. [DOI: 10.1016/j.asoc.2017.11.002] [Citation(s) in RCA: 20] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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10
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Majeed K, Masood Z, Samar R, Raja MAZ. A genetic algorithm optimized Morlet wavelet artificial neural network to study the dynamics of nonlinear Troesch’s system. Appl Soft Comput 2017. [DOI: 10.1016/j.asoc.2017.03.028] [Citation(s) in RCA: 33] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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11
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Raja MAZ, Shah FH, Alaidarous ES, Syam MI. Design of bio-inspired heuristic technique integrated with interior-point algorithm to analyze the dynamics of heartbeat model. Appl Soft Comput 2017. [DOI: 10.1016/j.asoc.2016.10.009] [Citation(s) in RCA: 70] [Impact Index Per Article: 10.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Raja MAZ, Mehmood A, Niazi SA, Shah SM. Computational intelligence methodology for the analysis of RC circuit modelled with nonlinear differential order system. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2806-6] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Raja MAZ, Kiani AK, Shehzad A, Zameer A. Memetic computing through bio-inspired heuristics integration with sequential quadratic programming for nonlinear systems arising in different physical models. SPRINGERPLUS 2016; 5:2063. [PMID: 27995040 PMCID: PMC5133222 DOI: 10.1186/s40064-016-3750-8] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/26/2016] [Accepted: 11/26/2016] [Indexed: 11/21/2022]
Abstract
Background In this study, bio-inspired computing is exploited for solving system of nonlinear equations using variants of genetic algorithms (GAs) as a tool for global search method hybrid with sequential quadratic programming (SQP) for efficient local search. The fitness function is constructed by defining the error function for systems of nonlinear equations in mean square sense. The design parameters of mathematical models are trained by exploiting the competency of GAs and refinement are carried out by viable SQP algorithm. Results Twelve versions of the memetic approach GA-SQP are designed by taking a different set of reproduction routines in the optimization process. Performance of proposed variants is evaluated on six numerical problems comprising of system of nonlinear equations arising in the interval arithmetic benchmark model, kinematics, neurophysiology, combustion and chemical equilibrium. Comparative studies of the proposed results in terms of accuracy, convergence and complexity are performed with the help of statistical performance indices to establish the worth of the schemes. Conclusions Accuracy and convergence of the memetic computing GA-SQP is found better in each case of the simulation study and effectiveness of the scheme is further established through results of statistics based on different performance indices for accuracy and complexity.
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Affiliation(s)
- Muhammad Asif Zahoor Raja
- Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock Campus, Attock, Pakistan
| | - Adiqa Kausar Kiani
- Department of Economics, Federal Urdu University of Arts Science and Technology, Islamabad, Pakistan
| | - Azam Shehzad
- Department of Mathematics, Preston University, Kohat, Islamabad Campus, Islamabad, Pakistan
| | - Aneela Zameer
- Department of Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, 45650 Pakistan
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Bio-inspired computational heuristics for parameter estimation of nonlinear Hammerstein controlled autoregressive system. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2677-x] [Citation(s) in RCA: 45] [Impact Index Per Article: 5.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Ahmad I, Raja MAZ, Bilal M, Ashraf F. Bio-inspired computational heuristics to study Lane-Emden systems arising in astrophysics model. SPRINGERPLUS 2016; 5:1866. [PMID: 27822440 PMCID: PMC5078133 DOI: 10.1186/s40064-016-3517-2] [Citation(s) in RCA: 60] [Impact Index Per Article: 7.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/08/2016] [Accepted: 10/11/2016] [Indexed: 12/03/2022]
Abstract
This study reports novel hybrid computational methods for the solutions of nonlinear singular Lane–Emden type differential equation arising in astrophysics models by exploiting the strength of unsupervised neural network models and stochastic optimization techniques. In the scheme the neural network, sub-part of large field called soft computing, is exploited for modelling of the equation in an unsupervised manner. The proposed approximated solutions of higher order ordinary differential equation are calculated with the weights of neural networks trained with genetic algorithm, and pattern search hybrid with sequential quadratic programming for rapid local convergence. The results of proposed solvers for solving the nonlinear singular systems are in good agreements with the standard solutions. Accuracy and convergence the design schemes are demonstrated by the results of statistical performance measures based on the sufficient large number of independent runs.
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Affiliation(s)
- Iftikhar Ahmad
- Department of Mathematics, University of Gujrat, Gujrat, 50700 Pakistan
| | - Muhammad Asif Zahoor Raja
- Department of Electrical Engineering, COMSATS Institute of Information Technology, Attock, 43600 Pakistan
| | - Muhammad Bilal
- Faculty of Science and Technology, University of Malaysia Pahang, Pekan, Pahang Malaysia
| | - Farooq Ashraf
- Department of Mathematics, University of Gujrat, Gujrat, 50700 Pakistan
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Raja MAZ, Zameer A, Khan AU, Wazwaz AM. A new numerical approach to solve Thomas-Fermi model of an atom using bio-inspired heuristics integrated with sequential quadratic programming. SPRINGERPLUS 2016; 5:1400. [PMID: 27610319 PMCID: PMC4994819 DOI: 10.1186/s40064-016-3093-5] [Citation(s) in RCA: 49] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 05/05/2016] [Accepted: 08/17/2016] [Indexed: 11/30/2022]
Abstract
In this study, a novel bio-inspired computing approach is developed to analyze the dynamics of nonlinear singular Thomas–Fermi equation (TFE) arising in potential and charge density models of an atom by exploiting the strength of finite difference scheme (FDS) for discretization and optimization through genetic algorithms (GAs) hybrid with sequential quadratic programming. The FDS procedures are used to transform the TFE differential equations into a system of nonlinear equations. A fitness function is constructed based on the residual error of constituent equations in the mean square sense and is formulated as the minimization problem. Optimization of parameters for the system is carried out with GAs, used as a tool for viable global search integrated with SQP algorithm for rapid refinement of the results. The design scheme is applied to solve TFE for five different scenarios by taking various step sizes and different input intervals. Comparison of the proposed results with the state of the art numerical and analytical solutions reveals that the worth of our scheme in terms of accuracy and convergence. The reliability and effectiveness of the proposed scheme are validated through consistently getting optimal values of statistical performance indices calculated for a sufficiently large number of independent runs to establish its significance.
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Affiliation(s)
| | - Aneela Zameer
- Department of Computer and Information Sciences, Pakistan Institute of Engineering and Applied Sciences (PIEAS), Nilore, Islamabad, 45650 Pakistan
| | - Aziz Ullah Khan
- Department of Basic Sciences, Riphah International University, Islamabad, Pakistan
| | - Abdul Majid Wazwaz
- Department of Mathematics, Saint Xavier University, Chicago, IL 60655 USA
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Design of artificial neural network models optimized with sequential quadratic programming to study the dynamics of nonlinear Troesch’s problem arising in plasma physics. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2530-2] [Citation(s) in RCA: 75] [Impact Index Per Article: 9.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Raja MAZ, Zameer A, Kiani AK, Shehzad A, Khan MAR. Nature-inspired computational intelligence integration with Nelder–Mead method to solve nonlinear benchmark models. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2523-1] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Solutions of Bagley–Torvik and Painlevé equations of fractional order using iterative reproducing kernel algorithm with error estimates. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2484-4] [Citation(s) in RCA: 41] [Impact Index Per Article: 5.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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20
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Stochastic numerical treatment for solving Falkner–Skan equations using feedforward neural networks. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2427-0] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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21
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Ahmad I, Raja MAZ, Bilal M, Ashraf F. Neural network methods to solve the Lane–Emden type equations arising in thermodynamic studies of the spherical gas cloud model. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2400-y] [Citation(s) in RCA: 88] [Impact Index Per Article: 11.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Raja MAZ, Shah FH, Khan AA, Khan NA. Design of bio-inspired computational intelligence technique for solving steady thin film flow of Johnson–Segalman fluid on vertical cylinder for drainage problems. J Taiwan Inst Chem Eng 2016. [DOI: 10.1016/j.jtice.2015.10.020] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Fateh MF, Zameer A, Mirza NM, Mirza SM, Raja MAZ. Biologically inspired computing framework for solving two-point boundary value problems using differential evolution. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2185-z] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Stochastic numerical solver for nanofluidic problems containing multi-walled carbon nanotubes. Appl Soft Comput 2016. [DOI: 10.1016/j.asoc.2015.10.015] [Citation(s) in RCA: 68] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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Raja MAZ, Khan JA, Chaudhary NI, Shivanian E. Reliable numerical treatment of nonlinear singular Flierl–Petviashivili equations for unbounded domain using ANN, GAs, and SQP. Appl Soft Comput 2016. [DOI: 10.1016/j.asoc.2015.10.017] [Citation(s) in RCA: 39] [Impact Index Per Article: 4.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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Design and application of nature inspired computing approach for nonlinear stiff oscillatory problems. Neural Comput Appl 2015. [DOI: 10.1007/s00521-015-1841-z] [Citation(s) in RCA: 34] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Raja MAZ, Khan JA, Shah SM, Samar R, Behloul D. Comparison of three unsupervised neural network models for first Painlevé Transcendent. Neural Comput Appl 2014. [DOI: 10.1007/s00521-014-1774-y] [Citation(s) in RCA: 26] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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