@Article{JCM-41-6,
author = {Jingrun, Chen and Shi, Jin and Lyu, Liyao},
title = {A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties},
journal = {Journal of Computational Mathematics},
year = {2023},
volume = {41},
number = {6},
pages = {1281--1304},
abstract = {<p style="text-align: justify;">We propose a deep learning based discontinuous Galerkin method (D2GM) to solve
hyperbolic equations with discontinuous solutions and random uncertainties. The main
computational challenges for such problems include discontinuities of the solutions and the
curse of dimensionality due to uncertainties. Deep learning techniques have been favored for
high-dimensional problems but face difficulties when the solution is not smooth, thus have
so far been mainly used for viscous hyperbolic system that admits only smooth solutions.
We alleviate this difficulty by setting up the loss function using discrete shock capturing
schemes – the discontinous Galerkin method as an example – since the solutions are smooth
in the discrete space. The convergence of D2GM is established via the Lax equivalence
theorem kind of argument. The high-dimensional random space is handled by the Monte-Carlo method. Such a setup makes the D2GM approximate high-dimensional functions
over the random space with satisfactory accuracy at reasonable cost. The D2GM is found
numerically to be first-order and second-order accurate for (stochastic) linear conservation
law with smooth solutions using piecewise constant and piecewise linear basis functions,
respectively. Numerous examples are given to verify the efficiency and the robustness
of D2GM with the dimensionality of random variables up to 200 for (stochastic) linear
conservation law and (stochastic) Burgers’ equation.</p>},
issn = {1991-7139},
doi = {https://doi.org/10.4208/jcm.2205-m2021-0277},
url = {https://global-sci.com/article/84164/a-deep-learning-based-discontinuous-galerkin-method-for-hyperbolic-equations-with-discontinuous-solutions-and-random-uncertainties}
}