Solving the linear transport equation by a deep neural network approach

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.

Adaptive neural network asymptotic tracking control for nonstrict feedback stochastic nonlinear systems

“…Besides, inspired by existing work [8,15] that have studied the convergence of the loss function and the neural network solution for the linear transport equation and Fokker-Planck equation. We show the convergence result based on the hypercoercivity analysis, that is uniform in the Knudsen number and owns an exponential decay in time.…”

Finite Difference Nets: A Deep Recurrent Framework for Solving Evolution PDEs

“…We mention that there are other meshfree methods to solve PDEs, such as by using evolutionary algorithms to search for the optimized solution, in particular elliptic type equations were studied in [17]. Besides, there have been recent works on using deep learning methods to study other types of equations, for example [2,13,14,29].…”