Multilayer perceptrons and radial basis function neural network methods for the solution of differential equations: A survey

Computers and Mathematics with Applications

Volumn 62, Issue 10, 2011, Pages 3796-3811

  Kumar, Manoj ^a   Yadav, Neha ^a  

^a MOTILAL NEHRU NATIONAL INSTITUTE OF TECHNOLOGY   (India)

Author keywords

Backpropagation algorithm; Differential equations; Multilayer perceptron; Neural network; Radial basis functions

Indexed keywords

MULTILAYER PERCEPTRON; RADIAL BASIS FUNCTION NEURAL NETWORKS; RADIAL BASIS FUNCTIONS; UNIVERSAL APPROXIMATION;

APPROXIMATION ALGORITHMS; BACKPROPAGATION ALGORITHMS; DIFFERENTIAL EQUATIONS; ELECTRIC LOADS; IMAGE SEGMENTATION; MULTILAYER NEURAL NETWORKS; MULTILAYERS; PATTERN RECOGNITION SYSTEMS; RESEARCH; SURVEYS;

RADIAL BASIS FUNCTION NETWORKS;

EID: 80055093807     PISSN: 08981221     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.camwa.2011.09.028     Document Type: Article

References (90)

1. A.R. Its, A.G. Izergin, V.E. Korepin, and N.A. Slavnov Differential equations for quantum correlation functions International Journal of Quantum Physics B 1990 1003 1037
2. A.V. Kotikov Differential equations method: the calculation of vertex-type Feynman diagrams Physics Letters B 259 1991 314 322
3. Y.Z. Peng Exact solutions for some nonlinear partial differential equations Physics Letters A 314 2003 401 408
4. J.G. Verwer, J.G. Blom, M.V. Loon, and E.J. Spee A comparison of stiff ODE solvers for atmospheric chemistry problems Atmospheric Environment 30 1996 49 58 (Pubitemid 3008885)
5. J. Behlke, and O. Ristau A new approximate whole boundary solution of the Lamm differential equation for the analysis of sedimentation velocity experiments Biophysical Chemistry 95 2002 59 68 (Pubitemid 34203676)
6. U. Salzner, P. Otto, and J. Ladik Numerical Solution of partial differential equation system describing chemical kinetics and diffusion in a cell with the aid of compartmentalization Journal of Computational Chemistry 11 1990 194 204
7. R.V. Culshaw, and S. Ruan A delay differential equation model of HIV infection of CD4+T-cells Mathematical Biosciences 165 2000 27 39 (Pubitemid 30305420)
8. G.A. Bocharov, and F.A. Rihan Numerical modeling in biosciences using delay differential equations Journal of Computational and Applied Mathematics 125 2000 183 199 (Pubitemid 32034874)
9. R. Noberg Differential equations for moments of present values in life insurance Mathematics and Economics 17 1995 171 180
10. J.C. Butcher The Numerical Analysis of Ordinary Differential Equations: Runge Kutta and General Linear Methods 1987 Wiley Interscience New York, NY, USA
11. J.G. Verwer Expicit RungeKutta methods for parabolic partial differential equations Applied Numerical Mathematics 22 1996 359 379 (Pubitemid 126410855)
12. J. Douglas, and B.F. Jones On predictorcorrector methods for nonlinear parabolic differential equations Journal of the Society for Industrial and Applied Mathematics 11 1963 195 204
13. R.W. Hamming Stable predictor corrector methods for ordinary differential equations Journal of the ACM 6 1959 37 47
14. M. Petravic, G. Kuo Petravic, and K.V. Roberts Automatic production of programmes for solving partial differential equations by finite difference methods Computer Physics Communications 4 1972 82 88
15. A.A. Tseng, and S.X. Gu A finite difference scheme with arbitrary mesh system for solving high order partial differential equations Computers & Structures 31 1989 319 328
16. B. Wu, and R.E. White One implementation variant of finite difference method for solving ODEs/DAEs Computers and Chemical Engineering 28 2004 303 309 (Pubitemid 38156256)
17. V.G. Korneev Iterative methods of solving system of equations of the finite element method USSR Computational Mathematics and Mathematical Physics 17 1977 109 129
18. M. Kumar, and P. Kumar Computational method for finding various solutions for a quasilinear elliptic equation of Kirchhoff type Advances in Engineering Software 40 2009 1104 1111
19. V. Thomee From finite differences to finite elements: a short history of numerical analysis of partial differential equations Journal of Computational and Applied Mathematics 128 2001 1 54 (Pubitemid 32248120)
20. M. Kumar, and P.K. Srivastava Computational techniques for solving differential equations by cubic, quintic and sextic spline International Journal for Computational Methods in Engineering Science & Mechanics 10 2009 108 115
21. S. Sallam, and W. Ameen Numerical solution of general nth order differential equations via splines Applied Numerical Mathematics 6 1990 225 238 (Pubitemid 20669496)
22. H.M. Ei-Hawary, and S.M. Mahmoud Spline collocation methods for solving delay differential equations Applied Mathematics and Computation 146 2003 359 372
23. M. Kumar, and Y. Gupta Methods for solving singular boundary value problems using splines: a survey Applied Mathematics and Computation 32 2010 265 278
24. N. Shawagfeh, and D. Kaya Comparing numerical methods for the solutions of systems of ordinary differential equations Applied Mathematics Letters 17 2004 323 328
25. J. Liu, and G. Hou Numerical solutions of the space- and time-fractional coupled Burgers equations by generalized differential transform method Applied Mathematics and Computation 217 2011 7001 7008
26. P. Darnia, and A. Ebadian A method for the numerical solution of the integro-differential equations Applied Mathematics and Computation 188 2007 657 668 (Pubitemid 46686756)
27. M. Kumar, and Parul Methods for solving singular perturbation problems arising in science and engineering Mathematical and Computer Modelling 54 2011 556 575
28. A. Saadatmandi, and M. Dehghan Numerical solution of higher order linear Fredholm integro-differential difference equation with variable coefficients Computers & Mathematics with Applications 59 2010 2996 3004
29. A. Tocino, and R. Ardanuy Runge Kutta methods for the numerical solution of stochastic differential equations Journal of Computational and Applied Mathematics 138 2002 219 241 (Pubitemid 34103342)
30. S. Abbasbandy, and A. Taati Numerical solution of nonlinear Volterra integro-differential equations with nonlinear differential part by the operational Tau method and error estimation Journal of Computational and Applied Mathematics 231 2009 106 113
31. M. Kumar, and N. Singh A collection of computational techniques for solving singular boundary value problems Advances in Engineering Software 40 2009 288 297
32. S. Momani, and K. Al-Khaled Numerical solutions for systems of fractional differential equations by the decomposition method Applied Mathematics and Computation 162 2005 1351 1365 (Pubitemid 40114500)
33. B.P. Sommeijer, P.J. Vander Houwen, and B. Neta Symmetric multistep methods for second-order differential equations with periodic solutions Applied Numerical Mathematics 2 1986 69 77
34. M. Ghasemi, M.T. Kajani, and E. Babolian Numerical solutions of the nonlinear integro differential equations: wavelet Galerkin method and homotopy perturbation method Applied Mathematics and Computation 188 2007 450 455 (Pubitemid 46705450)
35. M. Kumar, and P. Kumar Computational method for finding various solutions for a quasilinear elliptic equation of Kirchhoff type Advances in Engineering Software 40 2009 1104 1111
36. K. Maleknejad, and A. Arzhang Numerical solution of the Fredholm singular integro-differential equation with Cauchy kernel by using Taylor-series expansion and Galerkin method Applied Mathematics and Computation 182 2006 888 897 (Pubitemid 44781037)
37. M. Kumar, H.K. Mishra, and P. Singh A boundary value approach for singularly perturbed boundary value problems Advances in Engineering Software 40 2009 298 304
38. Q. Wang, and D. Cheng Numerical solution of damped nonlinear KleinGordon equations using variational method and finite element approach Applied Mathematics and Computation 162 2005 381 401
39. M. Kumar, and N. Singh Modified Adomian decomposition method and computer implementation for solving singular boundary value problems arising in various physical problems Computers and Chemical Engineering 34 2010 1750 1760
40. X. Yang, Y. Liu, and S. Bai A numerical solution of second order linear partial differential equation by differential transform Applied Mathematics and Computation 173 2006 792 802 (Pubitemid 43243538)
41. V.S. Erturk, and S. Momani Solving system of fractional differential equations using differential transform method Journal of Computational and Applied Mathematics 215 2008 142 151 (Pubitemid 351318110)
42. A.J. Meadre, and A.A. Fernandez The numerical solution of linear ordinary differential equations by feed forward neural networks Mathematical and Computer Modelling 19 1994 1 25
43. I.E. Lagaris, and A. Likas Artificial neural networks for solving ordinary and partial differential equations IEEE Transactions on Neural Networks 9 1998 987 1000 (Pubitemid 128743662)
44. S. He, K. Reif, and R. Unbehauen Multilayer neural networks for solving a class of partial differential equations Neural Networks 13 2000 385 396 (Pubitemid 30252014)
45. I.E. Lagaris, and A.C. Likas Neural network methods for boundary value problems with irregular boundaries IEEE Transactions on Neural Networks 11 5 2000 1041 1049
46. L.P. Aarts, and P.V. Veer Neural network method for solving partial differential equations Neural Processing Letters 14 2001 261 271 (Pubitemid 33110855)
47. H. Alli, A. Ucar, and Y. Demir The solution of vibration control problems using artificial neural networks Journal of the Franklin Institute 340 2003 307 325
48. N. Smaoui, and S. Al-Enezi Modelling the dynamics of nonlinear partial differential equations using neural networks Journal of Computational and Applied Mathematics 170 2004 27 58 (Pubitemid 38804593)
49. A. Malek, and R.S. Beidokhti Numerical solution for high order differential equations using a hybrid neural network-optimization method Applied Mathematics and Computation 183 2006 260 271 (Pubitemid 44908606)
50. H. Saxen, and F. Petterson Method for the selection of inputs and structure of feed forward neural networks Computers and Chemical Engineering 30 2006 1038 1045
51. X. Li-Ying, W. Hui, Z. Zhe-Zhao, The algorithm of neural networks on the initial value problems in ordinary differential equations, in: 2nd IEEE Conference on Industrial Electronics and Applications, 2007, pp. 813816.
52. Y. Shirvany, M. Hayati, and R. Moradian Numerical solution of the nonlinear Schrodinger equation by feedforward neural networks Communications in Nonlinear Science and Numerical Simulation 13 2008 2132 2145 (Pubitemid 351635964)
53. R.S. Beidokhti, and A. Malek Solving initial boundary value problems for system of partial differential equations using neural network and optimization techniques Journal of the Franklin Institute 346 2009 898 913
54. A.N. Kolmogrov On the representation of continuous functions of several variables by superpositions of continuous functions of one variable and addition American Mathematical Society Translations 28 1963 55 59
55. G. Cybenko Approximation of superpositions of a sigmoidal function Mathematics of Control, Signals, and Systems 2 1989 304 314
56. J.A. Nelder, and R. Mead A simplex method for function minimization The Computer Journal 7 1965 308 313
57. Y. Shirvany, M. Hayati, and R. Moradian Multilayer perceptron neural network with novel unsupervised training method for the solution of partial differential equations Applied Soft Computing 9 2009 20 29
58. I.G. Tsoulos, D. Gavrilis, and E. Glavas Solving differential equations with constructed neural networks Neurocomputing 72 2009 2385 2391
59. I. Tsoulos, D. Gavrilis, and E. Glavas Neural network construction and training using grammatical evolution Neurocomputing 72 2008 269 277
60. M. O'Neill, and C. Ryan Grammatical evolution IEEE Transactions on Evolutionary Computation 5 2001 349 358 (Pubitemid 32949966)
61. C. Filici Error estimation in the neural network solution of ordinary differential equations Neural Networks 23 2010 614 617
62. P.E. Zadunaisky On the estimation of errors propagated in the numerical integration of ordinary differential equations Numerische Mathematik 27 1976 21 39
63. P.E. Zadunaisky On the accuracy in the numerical solution of the N-body problem Celestial Mechanics 20 1979 209 230
64. S. Effati, and M. Pakdaman Artificial neural network approach for solving fuzzy differential equations Information Sciences 180 2010 1434 1457
65. N.K. Masmoudi, C. Rekik, M. Djemel, and N. Derbel Two coupled neural network based solution of the HamiltonJacobiBellman equation Applied Soft Computing 11 2011 2946 2963
66. V. Dua An artificial neural network approximation based decomposition approach for parameter estimation of system of ordinary differential equations Computers and Chemical Engineering 35 2011 545 553
67. N. Mai-Duy, and T. Tran-Cong Numerical solution of differential equations using multiquadric radial basis function networks Neural Networks 14 2001 185 199 (Pubitemid 32157010)
68. L. Jianyu, L. Siwei, Q. Yingjian, H. Yaping, Numerical solution of differential equations by radial basis function neural networks, in: Proceedings of International Joint Conference on Neural Networks, 2002, pp. 773777.
69. N. Mai-Duy, and T. Tran-Cong Mesh-free radial basis function network methods with domain decomposition for approximation of functions and numerical solution of Poisson's equation Engineering Analysis with Boundary Elements 26 2002 133 156 (Pubitemid 34040854)
70. N. Mai-Duy, and T. Tran-Cong Approximation of function and its derivatives using radial basis function networks Applied Mathematical Modelling 27 2003 197 220 (Pubitemid 36219627)
71. L. Jianyu, Luo Siwei, Qi Yingjian, Huang Yaping, Numerical solution of elliptic partial differential equation by growing radial basis function neural networks, in: Proceedings of the International Joint Conference on Neural Networks, 2003, pp. 8590.
72. E.J. Kansa, H. Power, G.E. Fasshauer, and L. Ling A volumetric integral radial basis function method for time dependent partial differential equations I formulation Engineering Analysis with Boundary Elements 28 2004 1191 1206
73. H. Zou, J. Lei, C. Pan, Design of a new kind of RBF neural network based on differential reconstruction, in: International Joint Conference on Neural Network and Brain, vol. 1, 2005, pp. 456460.
74. N. Mai-Duy Solving high order ordinary differential equations with radial basis function networks International Journal for Numerical Methods in Engineering 62 2005 824 852 (Pubitemid 40195781)
75. N. Mai-Duy, and T. Tran-Cong Solving biharmonic problems with scattered-point discretization using indirect radial basis function networks Engineering Analysis with Boundary Elements 30 2006 77 87 (Pubitemid 43120303)
76. A. Golbabai, and S. Seifollahi Radial basis function networks in the numerical solution of linear integro-differential equations Applied Mathematics and Computation 188 2007 427 432 (Pubitemid 46705448)
77. A. Golbabai, M. Mammadov, and S. Seifollahi Solving a system of nonlinear integral equations by an RBF network Computers and Mathematics with Applications 57 2009 1651 1658
78. N. Mai-Duy, and T. Tran-Cong A Cartesian-grid collocation method based on radial-basis-function networks for solving PDEs in irregular domains Numerical Methods for Partial Differential Equations 23 2007 1192 1210
79. A. Aminataei, and M.M. Mazarei Numerical solution of Poisson's equation using radial basis function networks on the polar coordinate Computers and Mathematics with Applications 56 2008 2887 2895
80. H. Chen, Li Kong, and W. Leng Numerical solution of PDEs via integrated radial basis function networks with adaptive training algorithm Applied Soft Computing 11 2011 855 860
81. S. Sarra Integrated radial basis functions based differential quadrature method and its performance Computers & Mathematics with Applications 43 2002 1283 1296
82. B. Choi, and J. Lee Comparison of generalizing ability on solving differential equations using backpropagation and reformulated radial basis function network Neurocomputing 73 2009 115 118
83. K. Valasoulis, D.L. Fotadis, I.E. Lagaris, A. Likas, Solving differential equations with neural networks: implementation on a DSP platform, in: 14th International Conference on Digital Signal Processing, vol. 2, 2002, pp. 12651268.
84. N. Vaziri, A. Hojabri, A. Erfani, M. Monsefi, and B. Nilforooshan Critical heat flux prediction by using radial basis function and multilayer perceptron neural networks: a comparison study Nuclear Engineering and Design 237 2007 377 385 (Pubitemid 46148982)
85. S.S. Chiddarwar, and N.R. Babu Comparison of RBF and MLP neural networks to solve inverse kinematic problem for 6R serial robot by a fusion approach Engineering Applications of Artificial Intelligence 23 2010 1083 1092
86. A.S. Yilmaz, and Z. Ozer Pitch angle control in wind turbines above the rated wind speed by multi-layer perceptron and radial basis function neural networks Expert Systems with Applications 36 2009 9767 9775
87. M. Kashaninejad, A.A. Dehghani, and M. Kashiri Modeling of wheat soaking using two artificial neural networks (MLP and RBF) Journal of Food Engineering 91 2009 602 607
88. B. Suchacz, and M. Wesoowski The recognition of similarities in trace elements content in medicinal plants using MLP and RBF neural networks Talanta 69 2006 37 42 (Pubitemid 43165499)
89. J.S. Chandnok, I.N. Kar, and S. Tuli Estimation of furnace exit gas temperature (FEGT) using optimized radial basis and back-propagation neural networks Energy Conversion and Management 49 2008 1989 1998
90. A. Afkhami, M.A. Tarighat, and M. Bahram Artificial neural networks for determination of enantiomeric composition of α-phenylglycine using UV spectra of cyclodextrin hostguest complexes: comparison of feed-forward and radial basis function networks Talanta 75 2008 91 98 (Pubitemid 351288754)