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Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networks

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Abstract

Applications of neural network algorithms have been grown in recent years and various architectures have been introduced by researchers for the purpose of solving different types of differential equations. Physics informed neural networks, functional link neural networks, and feed-forward differential equation neural networks are some of these architectures. In this paper, we introduce a new neural network for simulating the behavior of Emden–Fowler-type dynamic modeled as an ordinary/partial/system of differential equation, i.e. ODE/PDE/SDE which is based on the development of two introduced functional link neural network and feed-forward differential equation neural network for the partial/system of differential equations. This algorithm uses roots of shifted Chebyshev polynomials as a training data set and the The Levenberg–Marquardt algorithm is taken as an optimizer. To show the applicability of the proposed network, it is applied to some test problems and the obtained results are compared with some other neural network approaches and also, some other numerical algorithms. The reported results showed that the algorithm proposed in this paper is a powerful method for simulating the behavior of partial and system of differential equations and is more accurate than other methods in the literature.

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Acknowledgements

The corresponding author’s work as well as the work of the third author was supported by a grant from IPM (Grant number: 3-1399-18), and also by the Center of Excellence in Cognitive Neuropsychology (CECN). They sincerely thank IPM and CECN for their supports.

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Authors and Affiliations

  1. Department of Computer Sciences, Faculty of Mathematical Sciences, Shahid Beheshti University, Tehran, Iran

    M. Omidi, B. Arab & K. Parand

  2. Department of Cognitive Modeling, Institute for Cognitive and Brain Sciences, Shahid Beheshti University, Tehran, Iran

    A. H. Hadian Rasanan, J. A. Rad & K. Parand

  3. School of Computer Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran

    J. A. Rad

  4. Department of Statistics and Actuarial Science, University of Waterloo, Waterloo, Canada

    K. Parand

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  1. M. Omidi
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  2. B. Arab
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  3. A. H. Hadian Rasanan
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  5. K. Parand
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Correspondence to J. A. Rad.

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Omidi, M., Arab, B., Rasanan, A.H.H. et al. Learning nonlinear dynamics with behavior ordinary/partial/system of the differential equations: looking through the lens of orthogonal neural networks. Engineering with Computers 38 (Suppl 2), 1635–1654 (2022). https://doi.org/10.1007/s00366-021-01297-8

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