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A feedforward neural network framework for approximating the solutions to nonlinear ordinary differential equations

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Abstract

In this paper, we propose a method to approximate the solutions to nonlinear ordinary differential equations (ODE) using a deep learning feedforward artificial neural networks (ANNs). The efficiency of the proposed—unsupervised type machine learning—method is shown by solving two boundary value problems (BVPs) from quantum mechanics and nanofluid mechanics. The proposed mean-squared loss function is the sum of two terms: the first term satisfies the differential equation, while the second term satisfies the initial or boundary conditions. The total loss function is minimized by using general type of quasi-Newton optimization methods to get a desired network output. The approximation capability of the proposed method is verified for two sets of boundary value problems: first, a second-order nonlinear ODE and, second, a system of coupled nonlinear third-order ODEs. Point-wise comparison of our approximation shows a strong agreement with the available exact solutions and/or Runge–Kutta-based numerical solutions. We remark that the proposed algorithm minimizes the overall learnable network hyperparameters in a given initial or boundary value problems. More importantly, for the coupled system of third-order nonlinear ordinary differential equations, the proposed method does not need any adjustment with the initial/boundary conditions. Also, the current method does not require any special type of computational mesh. A straightforward minimization of total loss function yields a highly accurate results even with less number of epochs. Therefore, the proposed framework offers an attractive setting for the fluid mechanics community who are interested in studying heat and mass transfer problems.

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Acknowledgements

Authors would like to thank the support of Research and Innovation and College of Science & Engineering, Texas A &M University-Corpus Christi (TAMUCC) for this research. Also, authors like to acknowledge the high performance computing clusters at TAMUCC for providing access to the computing systems. The authors like to thank the anonymous reviewers for their valuable comments.

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Authors and Affiliations

  1. Department of Mathematics & Statistics, Texas A &M University-Corpus Christi, Corpus Christi, TX, 78412-5825, USA

    Pavithra Venkatachalapathy & S. M. Mallikarjunaiah

  2. Department of Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX, 79409-1042, USA

    Pavithra Venkatachalapathy

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  1. Pavithra Venkatachalapathy
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Correspondence to S. M. Mallikarjunaiah.

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Venkatachalapathy, P., Mallikarjunaiah, S.M. A feedforward neural network framework for approximating the solutions to nonlinear ordinary differential equations. Neural Comput & Applic 35, 1661–1673 (2023). https://doi.org/10.1007/s00521-022-07855-5

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