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Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning

Author

Listed:
  • Ian Holloway

    (Department of Mathematics, Air Force Institute of Technology, WPAFB, OH 45433, USA
    These authors contributed equally to this work.)

  • Aihua Wood

    (Department of Mathematics, Air Force Institute of Technology, WPAFB, OH 45433, USA
    These authors contributed equally to this work.)

  • Alexander Alekseenko

    (Department of Mathematics, California State University Northridge, Northridge, CA 91330, USA
    These authors contributed equally to this work.)

Abstract

The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for large problems. This paper aims to accelerate the computation of this integral via machine learning methods. In particular, we build a deep convolutional neural network to encode/decode the solution vector, and enforce conservation laws during post-processing of the collision integral before each time-step. Our preliminary results for the spatially homogeneous Boltzmann equation show a drastic reduction of computational cost. Specifically, our algorithm requires O ( n 3 ) operations, while asymptotically converging direct discretization algorithms require O ( n 6 ) , where n is the number of discrete velocity points in one velocity dimension. Our method demonstrated a speed up of 270 times compared to these methods while still maintaining reasonable accuracy.

Suggested Citation

  • Ian Holloway & Aihua Wood & Alexander Alekseenko, 2021. "Acceleration of Boltzmann Collision Integral Calculation Using Machine Learning," Mathematics, MDPI, vol. 9(12), pages 1-15, June.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:12:p:1384-:d:575189
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    References listed on IDEAS

    as
    1. Justin Sirignano & Konstantinos Spiliopoulos, 2017. "DGM: A deep learning algorithm for solving partial differential equations," Papers 1708.07469, arXiv.org, revised Sep 2018.
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