We introduce a hybrid filter that incorporates a mathematically accurate moment-based filter with a data driven filter for discontinuous Galerkin approximations to PDE solutions that contain discontinuities. Numerical solutions of PDEs suffer from an $\mathcal{O}(1)$ error in the neighborhood about discontinuities, especially for shock waves that arise in inviscid compressible flow problems. While post-processing filters, such as the Smoothness-Increasing Accuracy-Conserving (SIAC) filter, can improve the order of error in smooth regions, the $\mathcal{O}(1)$ error in the vicinity of a discontinuity remains. To alleviate this, we combine the SIAC filter with a data-driven filter based on Convolutional Neural Networks. The data-driven filter is specifically focused on improving the errors in discontinuous regions and therefore {\it only includes top-hat functions in the training dataset.} For both filters, a consistency constraint is enforced, while the SIAC filter additionally satisfies $r-$moments. This hybrid filter approach allows for maintaining the accuracy guaranteed by the theory in smooth regions while the hybrid SIAC-data-driven approach reduces the $\ell_2$ and $\ell_\infty$ errors about discontinuities. Thus, overall the global errors are reduced. We examine the performance of the hybrid filter about discontinuities for the one-dimensional Euler equations for the Lax, Sod, and sine-entropy (Shu-Osher) shock-tube problems.

翻译：我们提出了一种混合滤波器，它结合了数学上精确的基于矩的滤波器与数据驱动的滤波器，用于处理包含不连续性的偏微分方程解的间断伽辽金近似。偏微分方程的数值解在不连续性邻域内存在$\mathcal{O}(1)$量级的误差，特别是在无粘可压缩流动问题中产生的激波附近。虽然后处理滤波器，如平滑度递增精度保持（SIAC）滤波器，可以提高光滑区域的误差阶数，但不连续性附近的$\mathcal{O}(1)$误差依然存在。为了缓解这一问题，我们将SIAC滤波器与基于卷积神经网络的数据驱动滤波器相结合。数据驱动滤波器专门致力于改善不连续区域的误差，因此{\it 其训练数据集中仅包含顶帽函数。}对于这两种滤波器，我们均施加了一致性约束，而SIAC滤波器还额外满足$r-$矩条件。这种混合滤波器方法能够在光滑区域保持理论所保证的精度，同时混合SIAC-数据驱动方法降低了不连续性附近的$\ell_2$和$\ell_\infty$误差。因此，整体全局误差得以减小。我们针对Lax、Sod和正弦熵（Shu-Osher）激波管问题的一维欧拉方程，检验了该混合滤波器在不连续性附近的性能。