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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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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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