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Solving the linear transport equation by a deep neural network approach

  • * Corresponding author: Lin Mu

    * Corresponding author: Lin Mu

Liu is supported by the start-up fund by The Chinese University of Hong Kong

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  • In this paper, we study linear transport model by adopting deep learning method, in particular deep neural network (DNN) approach. While the interest of using DNN to study partial differential equations is arising, here we adapt it to study kinetic models, in particular the linear transport model. Moreover, theoretical analysis on the convergence of neural network and its approximated solution towards analytic solution is shown. We demonstrate the accuracy and effectiveness of the proposed DNN method in numerical experiments.

    Mathematics Subject Classification: Primary: 65N22, 68T07; Secondary: 65M60.

    Citation:

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  • Figure 1.  Structure of the deep neural network

    Figure 2.  Example 5.2.1 with $ \sigma_s = 0.0 $ and $ \sigma_a = 1.0 $: (a). exact solution $ \rho $; (b). DNN approximation $ \rho_h $

    Figure 3.  Example 5.2.1 with $ \sigma_s = 0.0,\sigma_a = 1.0 $: DNN approximation to $ \psi $ at time = 1.0

    Figure 4.  Example 5.2.1: Plot of DNN solutions for $ \rho_h $: (a) $ \sigma_s = 1 $ and $ \sigma_a = 0 $; (b) $ \sigma_s = 1.0 $ and $ \sigma_a = 1.0 $

    Figure 5.  Example 5.2.1 with $ \sigma_s = 1.0,\sigma_a = 0.0 $: DNN approximation to $ \psi $ at time = 1.0

    Figure 6.  Example 5.2.1 with $ \sigma_s = 1.0,\sigma_a = 1.0 $: DNN approximation to $ \psi $ at time = 1.0

    Figure 7.  Example 5.2.2: Plot of DNN approximation to $ \psi $ for $ \sigma_s = 1.0 $ and $ \sigma_a = 0.0 $ at different time

    Figure 9.  Example 5.2.2: Plots of DNN solutions $ \rho_h $ for $ k = 100 $ and (a) $ \sigma_s = 1.0 $, $ \sigma_a = 0.0 $; (b)$ \sigma_s = 1.0 $, $ \sigma_a = 1.0 $

    Figure 8.  Example 5.2.2: Plot of DNN approximation to $ \psi $ for $ \sigma_s = 1.0 $ and $ \sigma_a = 1.0 $ at different time

    Figure 10.  Example 5.2.2: Plots of angular average of discrete ordinate $ S_{100} $ solutions for $ k = 100 $: (a) $ \sigma_s = 1 $, $ \sigma_a = 0 $; (b) $ \sigma_s = 1 $, $ \sigma_a = 1 $

    Figure 11.  Example 5.2.3: The illustration of the boundary condition

    Figure 12.  Example 5.2.3: Case (1) $ \sigma_s = 1.0 $, $ \sigma_a = 9.0 $: (a). solution of $ \rho_h $ at different time; (b). 2-dimensional plot of DNN approximation to $ \psi $ on the $ x-\mu $ plane at $ t = 10 $

    Figure 13.  Example 5.2.3: Case (2) $ \sigma_s = 5.0 $, $ \sigma_a = 5.0 $: (a). solution of $ \rho_h $ at different time; (b). 2-dimensional plot of DNN approximation to $ \psi $ on the $ x-\mu $ plane at $ t = 10 $

    Figure 14.  Example 5.2.3: Case (3) $ \sigma_s = 9.0 $, $ \sigma_a = 1.0 $: (a). solution of $ \rho_h $ at different time; (b). 2-dimensional plot of DNN approximation to $ \psi $ on the $ x-\mu $ plane at $ t = 10 $

    Figure 15.  Example 5.2.3: Plots of DNN solutions of $ \rho_h $ at time = 10.0

    Table 1.  Example 5.2.1: Relative Errors in DNN approximations to $ \psi $

    time $ \sigma_s = 0.0,\sigma_a = 1.0 $ $ \sigma_s = 1.0,\sigma_a = 0.0 $ $ \sigma_s = 1.0,\sigma_a = 1.0 $
    0.0 2.3329e-4 2.7779e-4 2.8335e-4
    0.2 1.8372e-4 4.4767e-2 2.1264e-2
    0.4 1.5604e-4 4.1943e-2 2.7555e-2
    0.6 1.3525e-4 4.1276e-2 2.6886e-2
    0.8 1.1590e-4 4.7595e-2 2.8699e-2
    1.0 1.1543e-4 4.5483e-2 2.1431e-2
     | Show Table
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