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April  2022, 15(4): 669-686. doi: 10.3934/dcdss.2021070

## Solving the linear transport equation by a deep neural network approach

 1 Department of Mathematics, University of Massachusetts, Dartmouth, MA, 02747 2 Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong 3 Department of Mathematics, University of Georgia, Athens, GA 30602

* Corresponding author: Lin Mu

Received  January 2021 Revised  April 2021 Published  April 2022 Early access  June 2021

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

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.

Citation: Zheng Chen, Liu Liu, Lin Mu. Solving the linear transport equation by a deep neural network approach. Discrete and Continuous Dynamical Systems - S, 2022, 15 (4) : 669-686. doi: 10.3934/dcdss.2021070
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##### References:
Structure of the deep neural network
Example 5.2.1 with $\sigma_s = 0.0$ and $\sigma_a = 1.0$: (a). exact solution $\rho$; (b). DNN approximation $\rho_h$
Example 5.2.1 with $\sigma_s = 0.0,\sigma_a = 1.0$: DNN approximation to $\psi$ at time = 1.0
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 $\sigma_a = 1.0$
Example 5.2.1 with $\sigma_s = 1.0,\sigma_a = 0.0$: DNN approximation to $\psi$ at time = 1.0
Example 5.2.1 with $\sigma_s = 1.0,\sigma_a = 1.0$: DNN approximation to $\psi$ at time = 1.0
Example 5.2.2: Plot of DNN approximation to $\psi$ for $\sigma_s = 1.0$ and $\sigma_a = 0.0$ at different time
Example 5.2.2: Plots of DNN solutions $\rho_h$ for $k = 100$ and (a) $\sigma_s = 1.0$, $\sigma_a = 0.0$; (b)$\sigma_s = 1.0$, $\sigma_a = 1.0$
Example 5.2.2: Plot of DNN approximation to $\psi$ for $\sigma_s = 1.0$ and $\sigma_a = 1.0$ at different time
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) $\sigma_s = 1$, $\sigma_a = 1$
Example 5.2.3: The illustration of the boundary condition
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 $t = 10$
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 $t = 10$
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 $t = 10$
Example 5.2.3: Plots of DNN solutions of $\rho_h$ at time = 10.0
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
 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
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