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.
翻译:我们提出一种基于深度学习的不连续 Galerkin 方法(D2GM), 以解决具有不连续解决方案和随机不确定性的双曲方程式(D2GM) 。 这些问题的主要计算挑战包括解决方案的不连续性和由于不确定性而使维度受到诅咒。 深层学习技术被偏向于高层面问题,但在解决方案不顺利的情况下却面临困难, 因此迄今为止主要用于只承认平稳解决方案的超双曲系统。 我们通过使用离散休克捕捉方案设置损失函数来缓解这一困难。 我们通过使用离散空间的离散休克捕捉取方案- 分离的Galerkin 方法作为示例。 D2GM 的趋同性是通过Lax等量理论类辩论建立的。 高维度随机空间由蒙特-Carlo 方法处理。 这种设置使D2GM在随机空间上接近高维度功能,以合理的成本。 我们发现D2GM在数字上是第一阶和第二阶次精确的线性保护法, 以平滑度法的常态和直径直径直径直径直方函数为不同。