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    A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems

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    Ugail_et-al_Applied_Intelligence (2.062Mb)
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    Publication date
    2022-01
    Author
    Aggarwal, R.
    Ugail, Hassan
    Jha, R.K.
    Keyword
    Artificial neural networks
    Biharmonic equation
    Boundary value problems
    Von-Kármán equation
    Navier-Stokes equation
    Mesh free solutions
    Rights
    © The Author(s) 2021. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons. org/licenses/by/4.0/.
    Peer-Reviewed
    Yes
    Open Access status
    openAccess
    
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    Abstract
    Seeking efficient solutions to nonlinear boundary value problems is a crucial challenge in the mathematical modelling of many physical phenomena. A well-known example of this is solving the Biharmonic equation relating to numerous problems in fluid and solid mechanics. One must note that, in general, it is challenging to solve such boundary value problems due to the higher-order partial derivatives in the differential operators. An artificial neural network is thought to be an intelligent system that learns by example. Therefore, a well-posed mathematical problem can be solved using such a system. This paper describes a mesh free method based on a suitably crafted deep neural network architecture to solve a class of well-posed nonlinear boundary value problems. We show how a suitable deep neural network architecture can be constructed and trained to satisfy the associated differential operators and the boundary conditions of the nonlinear problem. To show the accuracy of our method, we have tested the solutions arising from our method against known solutions of selected boundary value problems, e.g., comparison of the solution of Biharmonic equation arising from our convolutional neural network subject to the chosen boundary conditions with the corresponding analytical/numerical solutions. Furthermore, we demonstrate the accuracy, efficiency, and applicability of our method by solving the well known thin plate problem and the Navier-Stokes equation.
    URI
    http://hdl.handle.net/10454/18843
    Version
    Published version
    Citation
    Aggarwal R, Ugail H and Jha RK (2022) A deep artificial neural network architecture for mesh free solutions of nonlinear boundary value problems. Applied Intelligence. 52: 916-926.
    Link to publisher’s version
    https://doi.org/10.1007/s10489-021-02474-4
    Type
    Article
    Collections
    Engineering and Informatics Publications

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