A Deep Learning Based Discontinuous Galerkin Method for Hyperbolic Equations with Discontinuous Solutions and Random Uncertainties (Journal of Computational Mathematics, 2023)
We propose a deep learning based discontinuous Galerkin method (D2GM) to solvehyperbolic equations with discontinuous solutions and random uncertainties. The maincomputational challenges for such problems include discontinuities of the solutions and thecurse of dimensionality due to uncertainties. Deep learning techniques have been favored forhigh-dimensional problems but face difficulties when the solution is not smooth, thus haveso 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 capturingschemes \u2013 the discontinous Galerkin method as an example \u2013 since the solutions are smoothin the discrete space. The convergence of D2GM is established via the Lax equivalencetheorem kind of argument. The high-dimensional random space is handled by the Monte-Carlo method. Such a setup makes the D2GM approximate high-dimensional functionsover the random space with satisfactory accuracy at reasonable cost. The D2GM is foundnumerically to be first-order and second-order accurate for (stochastic) linear conservationlaw with smooth solutions using piecewise constant and piecewise linear basis functions,respectively. Numerous examples are given to verify the efficiency and the robustnessof D2GM with the dimensionality of random variables up to 200 for (stochastic) linearconservation law and (stochastic) Burgers\u2019 equation.
Keywords: Discontinuous Galerkin method; Loss function; Convergence analysis; Deep learning; Hyperbolic equation.