Numerical solution of coupled system of Emden-Fowler equations using artificial neural network technique
In this paper, a deep artificial neural network technique is proposed to solve the coupled system of Emden-Fowler equations. A vectorized form of algorithm is developed. Implementation and simulation of this technique is performed using Python code. This technique is implemented in various numerical examples, and simulations are conducted. We have shown graphically how accurately this method works. We have shown the comparison of numerical solution and exact solution using error tables. We have also conducted a comparative analysis of our solution with alternative methods, including the Bernstein collocation method and the Homotopy analysis method. The comparative results are presented in error tables. The efficiency and accuracy of this method are demonstrated by these graphs and tables.
[1] Braun, M., Golubitsky, M. (1983). Differential Equations and Their Applications (Vol. 2). New York: Springer-Verlag.
[2] Simmons, G. F. (2016). Differential Equations with Applications and Historical Notes. CRC Press.
[3] Chandrasekhar, S., Chandrasekhar, S. (1957). An Introduction to the Study of Stellar Structure (Vol. 2). Courier Corporation.
[4] Lane, H. J. (1870). On the theoretical tempera- ture of the sun, under the hypothesis of a gaseous mass maintaining its volume by its internal heat, and depending on the laws of gases as known to terrestrial experiment. American Journal of Sci- ence, 2(148), 57-74.
[5] Fowler, R. H. (1914). Some results on the form near infinity of real continuous solutions of a cer- tain type of second order differential equation. Proceedings of the London Mathematical Society, 2(1), 341-371.
[6] Huang, J. T., Li, J., Gong, Y. (2015, April). An analysis of convolutional neural networks for speech recognition. In 2015 IEEE International Conference on Acoustics, Speech and Signal Pro- cessing (ICASSP) (pp. 4989-4993). IEEE.
[7] Seifert, C., Aamir, A., Balagopalan, A., Jain, D.,Sharma, A., Grottel, S., Gumhold, S. (2017). Vi- sualizations of deep neural networks in computer vision: A survey. Transparent Data Mining for Big and Small Data, 123-144.
[8] Dixit, P., Silakari, S. (2021). Deep learning algo- rithms for cybersecurity applications: A techno- logical and status review. Computer Science Re- view, 39, 100317.
[9] Vigneswaran, R. K., Vinayakumar, R., Soman, K. P., Poornachandran, P. (2018, July). Evalu- ating shallow and deep neural networks for net- work intrusion detection systems in cyber security. In 2018 9th International Conference on Comput- ing, Communication and Networking Technologies (ICCCNT) (pp. 1-6). IEEE.
[10] Szegedy, C., Zaremba, W., Sutskever, I., Bruna, J., Erhan, D., Goodfellow, I., Fergus, R. (2013). Intriguing properties of neural networks. arXiv preprint arXiv:1312.6199.
[11] Malladi, S., Sharapov, I. (2018). FastNorm: im- proving numerical stability of deep network train- ing with efficient normalization.
[12] Zheng, Z., Hong, P. (2018). Robust detection of adversarial attacks by modeling the intrinsic prop- erties of deep neural networks. Advances in Neural Information Processing Systems, 31.
[13] Haber, E., Ruthotto, L. (2017). Stable architec- tures for deep neural networks. Inverse Problems, 34(1), 014004.
[14] Balduzzi, D., Frean, M., Leary, L., Lewis, J. P., Ma, K. W. D., McWilliams, B. (2017, July). The shattered gradients problem: If resnets are the an- swer, then what is the question?. In International Conference on Machine Learning (pp. 342-350). PMLR.
[15] Liu, W., Wang, Z., Liu, X., Zeng, N., Liu, Y., Al- saadi, F. E. (2017). A survey of deep neural net- work architectures and their applications. Neuro- computing, 234, 11-26.
[16] Samek, W., Montavon, G., Lapuschkin, S., An- ders, C. J., M¨uller, K. R. (2021). Explaining deep neural networks and beyond: A review of methods and applications. Proceedings of the IEEE, 109(3), 247-278.
[17] Bj¨orck, A. (1990) Least squares methods. Hand- book of Numerical Analysis, Edited by P.G. Ciar- let and J.L. Lions, Volume 1, 465–652.
[18] Levenberg, K. (1944). A method for the solution of certain non-linear problems in least squares. Quarterly of applied mathematics, 2(2), 164-168.
[19] Fletcher, C. A., Fletcher, C. A. J. (1984). Com- putational Galerkin Methods. Springer Berlin Hei- delberg, pp. 72-85
[20] Lagaris, I. E., Likas, A., Fotiadis, D. I. (1998). Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks, 9(5), 987-1000.
[21] Tan, L. S., Zainuddin, Z., Ong, P. (2018, June). Solving ordinary differential equations using neu- ral networks. In AIP Conference Proceedings (Vol. 1974, No. 1). AIP Publishing.
[22] Michoski, C., Milosavljevi´c, M., Oliver, T., Hatch, D. R. (2020). Solving differential equations using deep neural networks. Neurocomputing, 399, 193- 212.
[23] Han, J., Jentzen, A., E, W. (2018). Solving high- dimensional partial differential equations using deep learning. Proceedings of the National Acad- emy of Sciences, 115(34), 8505-8510.
[24] Sabir, Z., Saoud, S., Raja, M. A. Z., Wahab, H. A., Arbi, A. (2020). Heuristic computing tech- nique for numerical solutions of nonlinear fourth order Emden–Fowler equation. Mathematics and Computers in Simulation, 178, 534-548.
[25] Sabir, Z., Raja, M. A. Z., Arbi, A., Altamirano, G. C., Cao, J. (2021). Neuro-swarms intelligent computing using Gudermannian kernel for solv- ing a class of second order Lane-Emden singular nonlinear model. AIMS Mathematics, 6(3), 2468- 2485.
[26] Sabir, Z., Raja, M. A. Z., Alhazmi, S. E., Gupta, M., Arbi, A., Baba, I. A. (2022). Applications of artificial neural network to solve the nonlinear COVID-19 mathematical model based on the dy- namics of SIQ. Journal of Taibah University for Science, 16(1), 874-884.
[27] Meade Jr, A. J., Fernandez, A. A. (1994). Solu- tion of nonlinear ordinary differential equations by feedforward neural networks. Mathematical and Computer Modelling, 20(9), 19-44.
[28] Aarts, L. P., Van Der Veer, P. (2001). Neural net- work method for solving partial differential equa- tions. Neural Processing Letters, 14, 261-271.
[29] Yazdi, H. S., Pakdaman, M., Modaghegh, H.(2011). Unsupervised kernel least mean square algorithm for solving ordinary differential equa- tions. Neurocomputing, 74(12-13), 2062-2071.
[30] Asady, B., Hakimzadegan, F., Nazarlue, R.(2014). Utilizing artificial neural network ap-proach for solving two-dimensional integral equa-tions.Mathematical Sciences,8, 1-9.
[31] Berg, J., Nystr ̈om, K. (2018). A unified deep arti-ficial neural network approach to partial differen-tial equations in complex geometries.Neurocom-puting,317, 28-41.
[32] Nascimento, R. G., Viana, F. A. (2020). Cumula-tive damage modeling with recurrent neural net-works.AIAA Journal,58(12), 5459-5471.
[33] Innes, M., Edelman, A., Fischer, K., Rackauckas,C., Saba, E., Shah, V. B., Tebbutt, W. (2019).A differentiable programming system to bridgemachine learning and scientific computing.arXivpreprint arXiv:1907.07587.
[34] Wang, H., Qin, C., Bai, Y., Zhang, Y.,Fu, Y. (2021). Recent advances on neural net-work pruning at initialization.arXiv preprintarXiv:2103.06460.
[35] Dufera, T. T. (2021). Deep neural network for sys-tem of ordinary differential equations: Vectorizedalgorithm and simulation.Machine Learning withApplications,5, 100058.
[36] Baydin, A. G., Pearlmutter, B. A., Radul, A. A.,Siskind, J. M. (2018). Automatic differentiation inmachine learning: a survey.Journal of MarchineLearning Research,18, 1-43.
[37] Shahni, J., Singh, R. (2021). Numerical solutionof system of Emden-Fowler type equations byBernstein collocation method.Journal of Math-ematical Chemistry,59(4), 1117-1138.
[38] Althubiti, S., Kumar, M., Goswami, P., Kumar,K. (2023). Artificial neural network for solvingthe nonlinear singular fractional differential equa-tions.Applied Mathematics in Science and Engi-neering,31(1), 2187389.
[39] Panghal, S., Kumar, M. (2022). Neural networkmethod: delay and system of delay differentialequations.Engineering with Computers,38(Suppl3), 2423-2432.
[40] Singh, R., Singh, G., Singh, M. (2021). Numer- ical algorithm for solution of the system of Em- den–Fowler type equations. International Journal of Applied and Computational Mathematics, 7(4), 136.
[41] Singh, R. (2018). Analytical approach for compu- tation of exact and analytic approximate solutions to the system of Lane-Emden-Fowler type equa- tions arising in astrophysics. The European Phys- ical Journal Plus, 133(8), 320.
[42] Wazwaz, A. M. (2011). The variational iteration method for solving systems of equations of Em- den–Fowler type. International Journal of Com- puter Mathematics, 88(16), 3406-3415.