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ANN-based deep collocation method for natural convection in porous media

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Abstract

A deep collocation method (DCM) is proposed for studying the natural convection phenomenon in the porous media (NCPM). The buoyancy-driven convection analysis inside the porous media is a complex process governed by dynamical conservation laws. Furthermore, these conservation laws involve complex nonlinear governing equations, which are required special computational techniques in well-known numerical methods like finite element method (FEM), finite difference method (FDM), finite volume method (FVM), and others. Such numerical schemes often face computational limitations like mesh generation, dimensionality limitations, increased computation errors for complex domains, and challenges in modeling physics. This research employs an unsupervised deep learning (DL) approach to address and resolve the typical computational challenges encountered in traditional numerical methods when dealing with natural convection in complex porous enclosures. In contrast to mesh-based numerical methods, the computational procedure in the DL approach involves domain and boundary discretization, followed by the random sampling of collocation points throughout the entire physical domain and its boundaries. Furthermore, a loss function is defined based on the governing differential equations and boundary conditions, which are minimized at the collocation points to achieve the desired solution. A combination of gradient-based optimizers is deployed to obtain a better set of parameter values using the backpropagation algorithm. The entire setup of the feedforward neural network is trained to approximate the solution with acceptable accuracy. The study explores four configurations of porous enclosures for a nonlinear mathematical model of natural convection in porous media, with various combinations of Neumann and Dirichlet boundary conditions. Additionally, the results from the mesh-based FEM are chosen as reference data to validate the consistency and accuracy of the DCM results. In all cases, the DCM results exhibit excellent agreement with the FEM results.

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Acknowledgements

The first author wishes to express sincere gratitude to D.I.A.T., Pune, for their financial support and academic guidance throughout his Ph.D. program. Also, he extends his sincere gratitude to the anonymous reviewers whose thoughtful feedback and constructive input have significantly enhanced the quality and depth of this research article.

Funding

The research conducted is part of fulfilling the Ph.D. degree requirements at the parent institution (DIAT) of the first author, and no external funds or financial support were secured for this research work.

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Authors and Affiliations

  1. Department of Applied Mathematics, Defence Institute of Advanced Technology, Pune, 411025, India

    Sumant Kumar & S. V. S. S. N. V. G. Krishna Murthy

  2. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, India

    B. V. Rathish Kumar

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  1. Sumant Kumar
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Contributions

SK contributed to conceptualization, formal analysis, investigation, methodology, software, validation, visualization, writing—original draft, and writing—review and editing. BVRK contributed to conceptualization, formal analysis, supervision, and writing—review and editing. SVSSNVGKM contributed to conceptualization, formal analysis, methodology, supervision, and writing—review and editing.

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Correspondence to S. V. S. S. N. V. G. Krishna Murthy.

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Kumar, S., Kumar, B.V.R. & Murthy, S.V.S.S.N.V.G.K. ANN-based deep collocation method for natural convection in porous media. Neural Comput & Applic 36, 6067–6083 (2024). https://doi.org/10.1007/s00521-023-09385-0

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