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

Abstract

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.

References in corpus (7)

• DGM: A deep learning algorithm for solving partial differential equations
• PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs
• Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems
• Weak Adversarial Networks for High-dimensional Partial Differential Equations
• Trend to Equilibrium for the Kinetic Fokker-Planck Equation via the Neural Network Approach
• MIM: A deep mixed residual method for solving high-order partial differential equations
• Reproducing Activation Function for Deep Learning