Using deep learning techniques for solving convection-dominated convection-diffusion equations

  • Frerichs-Mihov, Derk (WIAS)
  • Henning, Linus (Freie Universität Berlin)
  • John, Volker (WIAS and Freie Universität Berlin)

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Convection-diffusion equations are a basic model to describe the distribution of a scalar quantity in fluids. Besides modeling the heat distribution in a room, they describe the concentration of drugs in blood and the propagation of chemical substances in water. Unfortunately, many classical numerical methods fail in practical applications, namely in the so-called convection-dominated regime, i.e., when convection is much stronger than diffusion. In this regime the problem can be seen as a multiscale problem and these methods often produce unphysical values, so-called spurious oscillations [1, 2]. In the last two decades, the popularity of Deep Learning methods has risen sharply due to many success stories. In the field of numerical methods for solving partial differential equations so-called PINNs stand out, which try to integrate the underlying physical equations into the learning process [3]. They have an enormous potential and have been applied to various problems, see [4] and the references therein. The objective of this talk is to bring together deep learning methods and convection-dominated convection diffusion equations. It shows certain challenges and proposes ideas how to overcome them. The ideas are tested numerically for typical benchmark problems for convection-dominated convection-diffusion equations and are compared with classical methods. [1] Augustin, M., Caiazzo, A., John, V. et al. An assessment of discretizations for convection-dominated convection-diffusion equations. Computer Methods in Applied Mechanics and Engineering, 200(47-48), pp. 3395–3409, 2011, 10.1016/j.cma.2011.08.012 [2] Frerichs, D. and John, V. On reducing spurious oscillations in discontinuous Galerkin (DG) methods for steady-state convectiondiffusion equations. Journal of Computational and Applied Mathematics, 393, pp. 113487/1–113487/20, 2021, 10.1016/ [3] Raissi, M., Perdikaris, P. and Karniadakis, G. E. Physics Informed Deep Learning (Part I): Data- driven Solutions of Nonlinear Partial Differential Equations. ArXiv, arXiv:1711.10561v1, 2017, 10.48550/arXiv.1711.10561 [4] Karniadakis, G. E., Kevrekidis, I. G., Lu, L. et al. Physics-informed machine learning. Nature Reviews Physics, 3, pp. 422440, 2021, 10.1038/s42254-021-00314-5