A Novel Method for Solving Ordinary Differential Equations with Artificial Neural Networks

Show more

References

[1] McCulloch, W.S. and Pitts W. (1943) A Logical Calculus of the Ideas Immanent in Nervous Activity. The Bulletin of Mathematical Biophysics, 5, 115-133.

https://doi.org/10.1007/BF02478259

[2] Neumann, J.V. (1951) The General and Logical Theory of Automata. Wiley, New York.

[3] Graupe, D. (2007) Principles of Artificial Neural Networks. Vol. 6, 2nd Edition, World Scientific Publishing Co. Pte. Ltd., Singapore.

[4] Rumelhart, D.E. and McClelland, J.L. (1986) Parallel Distributed Processing, Explorations in the Microstructure of Cognition I and II. MIT Press, Cambridge.

https://doi.org/10.7551/mitpress/5236.001.0001

[5] Werbos, P.J. (1974) Beyond Recognition, New Tools for Prediction and Analysis in the Behavioural Sciences. Ph.D. Thesis, Harvard University, Cambridge.

[6] Lagaris, I.E., Likas, A.C. and Fotiadis, D.I. (1997) Artificial Neural Network for Solving Ordinary and Partial Differential Equations. arXiv: physics/9705023v1.

[7] Cybenco, G. (1989) Approximation by Superposition of a Sigmoidal Function. Mathematics of Control, Signals and Systems, 2, 303-314.

https://doi.org/10.1007/BF02551274

[8] Hornic, K., Stinchcombe, M. and White, H. (1989) Multilayer Feedforward Networks Are Universal Approximators. Neural Networks, 2, 359-366.

https://doi.org/10.1016/0893-6080(89)90020-8

[9] Lee, H. and Kang, I.S. (1990) Neural Algorithms for Solving Differential Equations. Journal of Computational Physics, 91, 110-131.

https://doi.org/10.1016/0021-9991(90)90007-N

[10] Majidzadeh, K. (2011) Inverse Problem with Respect to Domain and Artificial Neural Network Algorithm for the Solution. Mathematical Problems in Engineering, 2011, Article ID: 145608, 16 p.

https://doi.org/10.1155/2011/145608

[11] Chen, R.T.Q., Rubanova, Y., Bettencourt, J. and Duvenaud, D. (2018) Neural Ordinary Differential Equations. arXiv: 1806.07366v1.

[12] Okereke, R.N. (2019) A New Perspective to the Solution of Ordinary Differential Equations using Artificial Neural Networks. Ph.D Dissertation, Mathematics Department, Michael Okpara University of Agriculture, Umudike.

[13] Mall, S. and Chakraverty, S. (2013) Comparison of Artificial Neural Network Architecture in Solving Ordinary Differential Equations. Advances in Artificial Neural Systems, 2013, Article ID: 181895.

https://doi.org/10.1155/2013/181895

[14] IBM (2015) IBM SPSS Statistics 23

http://www.ibm.com

[15] PTC (Parametric Technology Corporation) (2007) Mathcad Version 14.

http://communications@ptc.com

[16] Otadi, M. and Mosleh, M. (2011) Numerical Solution of Quadratic Riccati Differential Equations by Neural Network. Mathematical Sciences, 5, 249-257.