  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.
