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We propose a deep learning based discontinuous Galerkin method (D2GM) tosolve hyperbolic equations with discontinuous solutions and randomuncertainties. The main computational challenges for such problems includediscontinuities of the solutions and the curse of dimensionality due touncertainties. Deep learning techniques have been favored for high-dimensionalproblems but face difficulties when the solution is not smooth, thus have sofar been mainly used for viscous hyperbolic system that admits only smoothsolutions. We alleviate this difficulty by setting up the loss function usingdiscrete shock capturing schemes--the discontinous Galerkin method as anexample--since the solutions are smooth in the discrete space. The convergenceof D2GM is established via the Lax equivalence theorem kind of argument. Thehigh-dimensional random space is handled by the Monte-Carlo method. Such asetup makes the D2GM approximate high-dimensional functions over the randomspace with satisfactory accuracy at reasonable cost. The D2GM is foundnumerically to be first-order and second-order accurate for (stochastic) linearconservation law with smooth solutions using piecewise constant and piecewiselinear basis functions, respectively. Numerous examples are given to verify theefficiency and the robustness of D2GM with the dimensionality of randomvariables up to $200$ for (stochastic) linear conservation law and (stochastic)Burgers' equation.

A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties