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Solving ordinary differential equations using an optimization technique based on training improved artificial neural networks

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Abstract

The solution of ordinary differential equations (ODEs) arises in a wide variety of engineering problems. This paper presents a novel method for the numerical solution of ODEs using improved artificial neural networks (IANNs). In the first step, we derive an approximate solution of ODEs by artificial neural networks (ANNs). Then, we construct a joint cost function of network system, it consists of several error functions corresponding to different sample points, and we reformulate Levenberg–Marquardt (RLM) algorithm to adjust the network parameters. The advantages of this method are high calculation accuracy and fast convergence speed compared with other existed methods, also increasing the simulation stability of ANNs method. The performance of the new proposed method in terms of calculation accuracy and convergence speed is analyzed for several different types of nonlinear ODEs.

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References

  • Attili BS (2010) The Hilber–Hughes–Taylor-α (HHT-α) method compared with an implicit Runge–Kutta for second-order systems. Int J Comput Math 87(8):1755–1767

    Article  MathSciNet  Google Scholar 

  • Boyd JP (2011) One-point pseudospectral collocation for the one-dimensional Bratu equation. Appl Math Comput 217(12):5553–5565

    MathSciNet  MATH  Google Scholar 

  • Chakraverty S, Mall S (2014) Regression-based weight generation algorithm in neural network for solution of initial and boundary value problems. Neural Comput Appl 25(3):585–594

    Article  Google Scholar 

  • Deeba E, Khuri SA, Xie S (2000) An algorithm for solving boundary value problems. J Comput Phys 159(2):125–138

    Article  MathSciNet  Google Scholar 

  • Douglas J, Jones BF (1963) On predictor–corrector methods for nonlinear parabolic differential equations. J Soc Ind Appl Math 11(1):195–204

    Article  MathSciNet  Google Scholar 

  • Effati S, Pakdaman M (2010) Artificial neural network approach for solving fuzzy differential equations. Inf Sci 180(8):1434–1457

    Article  MathSciNet  Google Scholar 

  • Feng X, Mei L, He G (2007) An efficient algorithm for solving Troesch’s problem. Appl Math Comput 189(1):500–507

    MathSciNet  MATH  Google Scholar 

  • Ha SN (2001) A nonlinear shooting method for two-point boundary value problems. Comput Math Appl 42(11–12):1411–1420

    Article  MathSciNet  Google Scholar 

  • Hagan MT, Menhaj MB (1994) Training feed-forward networks with the Marquardt algorithm. IEEE Trans Neural Netw 5(6):989–993

    Article  Google Scholar 

  • Hoda SA, Nagla HA (2011) Neural network methods for mixed boundary value problems. Int J Nonlinear Sci 11(3):312–316

    MathSciNet  Google Scholar 

  • Hornik K (1989) Multilayer feedforward networks are universal approximators. Neural Netw 2(5):359–366

    Article  Google Scholar 

  • Hou C, Simos TE, Famelis IT (2020) Neural network solution of pantograph type differential equations. Math Methods Appl Sci 43(6):3369–3374

    Article  MathSciNet  Google Scholar 

  • Lagaris IE, Likas A, Fotiadis DI (1998) Artificial neural networks for solving ordinary and partial differential equations. IEEE Trans Neural Netw 9(5):987–1000

    Article  Google Scholar 

  • Lagaris IE, Likas A, Papageorgiou DG (2000) Neural-network methods for boundary value problems with irregular boundaries. IEEE Trans Neural Netw 11(5):1041–1049

    Article  Google Scholar 

  • Lee H, Kang IS (1990) Neural algorithm for solving differential equations. J Comput Phys 91(1):110–131

    Article  MathSciNet  Google Scholar 

  • Lu Y, Yin Q, Li H, Sun H, Yang Y, Hou M (2019) The LS-SVM algorithms for boundary value problems of high-order ordinary differential equations. Adv Differ Equ 1:1–22

    MathSciNet  MATH  Google Scholar 

  • Malek A, Beidokhti RS (2006) Numerical solution for high order differential equations using a hybrid neural network-optimization method. Appl Math Comput 183(1):260–271

    MathSciNet  MATH  Google Scholar 

  • Mall S, Chakraverty S (2013) Regression-based neural network training for the solution of ordinary differential equations. Int J Math Model Numer Optim 4(2):136–149

    MATH  Google Scholar 

  • Mall S, Chakraverty S (2016) Application of legendre neural network for solving ordinary differential equations. Appl Soft Comput 43:347–356

    Article  Google Scholar 

  • Mcfall K, Mahan JR (2009) Artificial neural network method for solution of boundary value problems with exact satisfaction of arbitrary boundary conditions. IEEE Trans Neural Netw 20(8):1221–1233

    Article  Google Scholar 

  • Mehrkanoon S, Falck T, Suykens JA (2012) Approximate solutions to ordinary differential equations using least squares support vector machines. IEEE Trans Neural Netw 23(9):1356–1367

    Article  Google Scholar 

  • Mickens RE (1994) Nonstandard finite difference models of differential equations. World Scientific, Singapore

    MATH  Google Scholar 

  • Momani S, Abuasad S, Odibat Z (2006) Variational iteration method for solving nonlinear boundary value problems. Appl Math Comput 183(2):1351–1358

    MathSciNet  MATH  Google Scholar 

  • Mosleh M (2013) Fuzzy neural network for solving a system of fuzzy differential equations. Appl Soft Comput 13(8):3597–3607

    Article  Google Scholar 

  • Otadi M (2019) Simulation and evaluation of second-order fuzzy boundary value problems. Soft Comput 23(20):10463–10475

    Article  Google Scholar 

  • Panghal S, Kumar M (2020) Optimization free neural network approach for solving ordinary and partial differential equations. Eng Comput. https://doi.org/10.1007/s00366-020-00985-1

    Article  Google Scholar 

  • Ricardo H (2009) A modern introduction to differential equations. Academic Press, Elsevier

    MATH  Google Scholar 

  • Rizaner FB, Rizaner A (2018) Approximate solutions of initial value problems for ordinary differential equations using radial basis function networks. Neural Process Lett 48(2):1063–1071

    Article  Google Scholar 

  • Sneddon IN (2006) Elements of partial differential equations. Courier Corporation, Chelmsford

    MATH  Google Scholar 

  • Troesch BA (1976) A simple approach to a sensitive two-point boundary value problem. J Comput Phys 21(3):279–290

    Article  MathSciNet  Google Scholar 

  • Wambecq A (1978) Rational Runge–Kutta methods for solving systems of ordinary differential equations. Computing 20(4):333–342

    Article  MathSciNet  Google Scholar 

  • Yang Y, Hou M, Luo J (2018) A novel improved extreme learning machine algorithm in solving ordinary differential equations by Legendre neural network methods. Adv Diff Equ 1:1–24

    MathSciNet  MATH  Google Scholar 

  • Yang Y, Hou M, Luo J, Tian Z (2020) Numerical solution of several kinds of differential equations using block neural network method with improved extreme learning machine algorithm. J Intell Fuzzy Syst 38(3):3445–3461

    Article  Google Scholar 

  • Yazdi HS, Pourreza R (2010) Unsupervised adaptive neural-fuzzy inference system for solving differential equations. Appl Soft Comput 10(1):267–275

    Article  Google Scholar 

  • Yazdi HS, Pakdaman M, Modaghegh H (2011) Unsupervised kernel least mean square algorithm for solving ordinary differential equations. Neurocomputing 74(12):2062–2071

    Article  Google Scholar 

  • Yazdi HS, Modaghegh H, Pakdaman M (2012) Ordinary differential equations solution in kernel space. Neural Comput Appl 21(1):79–85

    Article  Google Scholar 

  • Yildirim A, Ozis T (2009) Solutions of singular IVPs of Lane–Emden type by the variational iteration method. Nonlinear Anal theory Methods Appl 70(6):2480–2484

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant No. 51475086), CAST-BISEE2019-019, and the Fundamental Research Funds for the Central Universities (Grant No. N162312001).

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Authors and Affiliations

  1. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Shangjie Li

  2. Research Center of Mechanical Kinetics and Reliability, Northeastern University, Qinhuangdao, 066004, China

    Xingang Wang

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  1. Shangjie Li
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  2. Xingang Wang
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Correspondence to Xingang Wang.

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Author Shangjie Li declares that he has no conflict of interest. Author Xingang Wang declares that he has no conflict of interest.

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Li, S., Wang, X. Solving ordinary differential equations using an optimization technique based on training improved artificial neural networks. Soft Comput 25, 3713–3723 (2021). https://doi.org/10.1007/s00500-020-05401-w

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