Solving the linear transport equation by a deep neural network approach

Zheng Chen; Liu Liu; Lin Mu

doi:10.3934/dcdss.2021070

Article Contents

2022, Volume 15, Issue 4: 669-686. Doi: 10.3934/dcdss.2021070

This issue Previous Article A dictionary learning algorithm for compression and reconstruction of streaming data in preset order Next Article Stable numerical methods for a stochastic nonlinear Schrödinger equation with linear multiplicative noise

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
^* Corresponding author: Lin Mu

Received: December 31, 2020

Revised: March 31, 2021

Early access: June 2021

Published: April 2022

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

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Abstract

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.

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

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Figure 1. Structure of the deep neural network

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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 $

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

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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 $

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

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

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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

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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 $

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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

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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 $

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

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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 $

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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 $

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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 $

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

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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

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