Solving the linear transport equation by a deep neural network approach

TL;DR

This paper introduces a deep neural network approach to solve the linear transport equation, demonstrating its accuracy and convergence through theoretical analysis and numerical experiments.

Contribution

It adapts deep neural networks to kinetic models, specifically the linear transport equation, and provides convergence analysis and numerical validation.

Findings

DNN method accurately solves the linear transport equation

Theoretical convergence of the neural network solution is established

Numerical experiments confirm the effectiveness of the approach

Abstract

In this paper, we study the linear transport model by adopting the deep learning method, in particular the 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 the neural network and its approximated solution towards the analytic solution is shown. We demonstrate the accuracy and effectiveness of the proposed DNN method in the numerical experiments.