Design optimal neural network based on new LM training algorithm for solving 3D - PDEs

In this article, we design an optimal neural network based on new LM training algorithm. The traditional algorithm of LM required high memory, storage and computational overhead because of it required the updated of Hessian approximations in each iteration. The suggested design implemented to converts the original problem into a minimization problem using feed forward type to solve non-linear 3D - PDEs. Also, optimal design is obtained by computing the parameters of learning with highly precise. Examples are provided to portray the efficiency and applicability of this technique. Comparisons with other designs are also conducted to demonstrate the accuracy of the proposed design.
[1] Salih, H., Tawfiq, L. N. M., Yahya, Z. R., & Zin, S. M. (2018). Solving modified regularized long wave equation using collocation method. Journal of Physics: Conference Series, 1003(1), 012062. https://doi.org/10.1088/1742-6596/1003/1 /012062
[2] Hussein, N.A., & Tawfiq L. N.M. (2023). Exact soliton solution for systems of non-linear (2+1)D- DEs. AIP Conference Proceedings, 2834(1), 1-7.
[3] Jabber, A. K., & Tawfiq, L. N. M. (2018). New transform fundamental properties and its appli- cations. Ibn Alhaitham Journal for Pure and Ap- plied Science, 31(1), 151-163. https://doi.org/ 10.30526/31.2.1954
[4] Ali, S., Khan, A., Shah, K., Alqudah, M. A., & Abdeljawad, T. (2022). On computational anal- ysis of highly nonlinear model addressing real world applications. Results in Physics, 36, 105431. https://doi.org/10.1016/j.rinp.2022.1054 31
[5] Gul, H., Alrabaiah, H., Ali, S., Shah, K., & Muhammad, S. (2020). Computation of solution to fractional order partial reaction diffusion equa- tions. Journal of Advanced Research, 25, 31-38. ht tps://doi.org/10.1016/j.jare.2020.04.021
[6] Tawfiq, L. N., & Hussein, N. A. (2023). Efficient approach for solving (2+ 1) D-differential equa- tions. Baghdad Science Journal, 20(1), 0166-0166. https://doi.org/10.21123/bsj.2022.6541
[7] Enadi, M. O., & Tawfiq, L. N. M. (2019). New approach for solving three dimensional space par- tial differential equation. Baghdad Science Jour- nal, 16(3), 786-792. https://doi.org/10.21123 /bsj.2019.16.3(Suppl.).0786
[8] Tawfiq, L. N. M., & Altaie, H. (2020). Recent modification of homotopy perturbation method for solving system of third order PDEs. Jour- nal of Physics: Conference Series, 1530(1), 1-7. https://doi.org/10.1088/1742-6596/1530/1 /012073
[9] Ghazi, F. F. (2020). Modeling the contamination of soil adjacent to Mohammed AL-Qassim high- way in Baghdad. Iraqi Journal of Science, 61(10), 2663-2670. https://doi.org/10.24996/ijs.2020.61.10.23
[10] Tawfiq, L. N. M., & Kareem, Z. H. (2021). Effi- cient modification of the decomposition method for solving a system of PDEs. Iraqi Journal of Science, 62(9), 3061-3070. https://doi.org/10 .24996/ijs.2021.62.9.21
[11] Kareem, Z. H., & Tawfiq, L. N. M. (2020). Re- cent modification of decomposition method for solving nonlinear partial differential equations. Journal of Advances in mathematics, 18, 154-161. https://doi.org/10.24297/jam.v18i.8744
[12] Kareem, Z. H., & Tawfiq, L. N. M. (2023). Re- cent modification of decomposition method for solving wave-like Equation. Journal of Interdis- ciplinary Mathematics, 26(5), 809-820. https: //doi.org/10.47974/JIM-1235
[13] Tawfiq, L. N., & Hussein, N. A. (2022). Ex- act solution for systems of nonlinear (2+ 1) D- differential equations. Iraqi Journal of Science, 63(10), 4388-4396. https://doi.org/10.24996 /ijs.2022.63.10.25
[14] Tawfiq, L. N. M., & Abed, A. I. (2021). Effi- cient method for solving fourth order PDEs. Jour- nal of Physics: Conference Series, 1818(1), 1-10. https://doi.org/10.1088/1742-6596/1818/1 /012166
[15] Kareem, Z.H., & Tawfiq, L. N.M. (2022). New modification of decomposition method for solving high order strongly nonlinear partial differential equations. AIP Conference Proceedings, 2398(1), 1-9.
[16] Hussein, N. A., & Tawfiq, L. N. M. (2020, May). New approach for solving (1+ 1)-dimensional dif- ferential equation. Journal of Physics: Confer- ence Series, 1530(1), 1-11. https://doi.org/ 10.1088/1742-6596/1530/1/012098
[17] Tawfiq, L. N., & Yassien, S. M. (2013). Solution of high order ordinary boundary value problems using semi-analytic technique. Ibn Al-Haitham Journal for Pure & Applied Sciences, 26(1), 281- 291.
[18] Hussein, N.A., & Tawfiq, L.N.M. (2022). Effi- cient approach for solving high order (2+1) D- differential equation. AIP Conference Proceed- ings, 2398(1), pp. 1-11. https://doi.org/10 .1063/5.0093671
[19] Salih, H., & Tawfiq, L. (2020, November). So- lution of modified equal width equation using quartic trigonometric-spline method. Journal of Physics: Conference Series, 1664(1), 1-10. http s://doi.org/10.1088/1742-6596/1664/1/012 033
[20] Tawfiq, L. N. M., & Khamas, A. H. (2020, May). New coupled method for solving Burger’s equation. Journal of Physics: Conference Series, 1530(1), 1-11. https://doi.org/10.1088/1742 -6596/1530/1/012069
[21] Tawfiq, L. N. M., & Khamas, A. H. (2023). New approach for calculate exponential integral func- tion. Iraqi Journal of Science, 64(8), 4034-4042. https://doi.org/10.24996/ijs.2023.64.8.2 7
[22] Tawfiq, L. N. M., Al-Noor, N. H., & Al-Noor, T. H. (2019, September). Estimate the rate of con- tamination in baghdad soils by using numerical method. Journal of Physics: Conference Series, 1294(3), 1-11. https://doi.org/10.1088/1742 -6596/1294/3/032020
[23] Tawfiq, L. N., & Oraibi, Y. A. (2017). Fast train- ing algorithms for feed forward neural networks. Ibn Al-Haitham Journal for Pure and Applied Sci- ence, 26(1), 275-280.
[24] Tawfiq, L. N., & Hussein, A. A. (2013). De- sign feed forward neural network to solve singular boundary value problems. International Scholarly Research Notices, 2013, 1-7. https://doi.org/ 10.1155/2013/650467
[25] Tawfiq, L. N. M., & Hussein, W. R. (2016). Design suitable neural network for processing face recog- nition. Global Journal of Engineering Science and Researches, 3(3), 58-64.
[26] Tawfiq, L. N. M. (2017). The finite element neu- ral network and its applications to forward and inverse problems. Ibn AL-Haitham Journal For Pure and Applied Science, 19(4), 109-124.
[27] Tawfiq, L. N. M., & Salih, O. M. (2019). De- sign suitable feed forward neural network to solve Troesch’s problem. Sci. Int.(Lahore), 31(1), 41- 48.
[28] Hussien, Z. (2020). Anomaly detection approach based on deep neural network and dropout. Bagh- dad Science Journal, 17(2 (SI)), 0701-0701. http s://doi.org/10.21123/bsj.2020.17.2(SI). 0701
[29] Ali, M. H., & Tawfiq, L. N. (2023). Design optimal neural network for solving unsteady state con- fined aquifer problem. Mathematical Modelling of Engineering Problems, 10(2), 565-571. https: //doi.org/10.18280/mmep.100225
[30] Alia, M. H., & Tawfiqa, L. N. (2023). Novel neu- ral network based on New modification of BFGS update algorithm for solving partial differential equations. Advances in the Theory of Nonlinear Analysis and its Applications, 7(4), 76-88.
[31] Gupta, R., & Batra, C. M. (2022). Performance assessment of solar-transformer-consumption sys- tem using neural network approach. Baghdad Sci- ence Journal, 19(4), 0865-0865. https://doi.or g/10.21123/bsj.2022.19.4.0865
[32] Tawfiq, L. N. M., & Khamas, A. H. (2021). De- termine the effect hookah smoking on health with different types of tobacco by using parallel pro- cessing technique. Journal of Physics: Conference Series, 1818(1), 1-10. https://doi.org/10.108 8/1742-6596/1818/1/01217
[33] Tawfiq, L. N. M., & Tawfiq, M. N. M. (2017). The effect of number of training samples for ar- tificial neural network. Ibn AL-Haitham Journal For Pure and Applied Science, 23(3), 1-7.
[34] Ghazi, F. F., & Tawfiq, L. N. M. (2020). New ap- proach for solving two dimensional spaces PDE. Journal of Physics: Conference Series, 1530(1),012066. https://doi.org/10.1088/1742-659 6/1530/1/012066
[35] Jamil, H.J., Albahri, M.R.A., Al-Noor, N.H., Al- Noor, T.H., Heydari, A.R., Rajan, A.K., Arnetz, J., Arnetz, B. & Tawfiq, L.N.M. (2020). Hookah smoking with health risk perception of different types of tobacco. Journal of Physics: Conference Series, 1664(1), 012127. https://doi.org/10.1 088/1742-6596/1664/1/012127
[36] Kareema, Z. H., & Tawfiqa, L. N. (2023). Solv- ing (3+ 1) D-New Hirota bilinear equation using tanh method and new modification of extended tanh method. Advances in the Theory of Nonlin- ear Analysis and its Applications, 7(4), 114-122.
[37] Hussein, N. A., Helal, M. M., & Tawfiq, L. N. M.(2023). Double LA-transform and their proper- ties for solving partial differential equations. AIP Conference Proceedings, 2834(1), 1-10
[38] Kumar, A., Kumar, M., & Goswami, P. (2024). Numerical solution of coupled system of Emden- Fowler equations using artificial neural network technique. An International Journal of Optimiza- tion and Control: Theories & Applications, 14(1), 62-73. https://doi.org/10.11121/ijocta.14 24
[39] Okkan, U. (2011). Application of Levenberg- Marquardt optimization algorithm based multi- layer neural networks for hydrological time series modeling. An International Journal of Optimiza- tion and Control: Theories & Applications, 1(1), 53-63. https://doi.org/10.11121/ijocta.01 .2011.0038
[40] Kumar, K., Parida, M., & Katiyar, V. K. (2011). Road traffic noise prediction with neural networks- A review. An International Journal of Optimiza- tion and Control: Theories & Applications, 2(1), 29-37. https://doi.org/10.11121/ijocta.01 .2012.0059
[41] Demirtas, M., & Alci, M. (2011). A compara- tive study of neural networks and fuzzy systems in modeling of a nonlinear dynamic system. An International Journal of Optimization and Con- trol: Theories & Applications, 1(1), 65-73. https: //doi.org/10.11121/ijocta.01.2011.0055