The growing computing power over the years has enabled simulations to become more complex and accurate. However, high-fidelity simulations, while immensely valuable for scientific discovery and problem solving, come with significant computational demands. As a result, it is common to run a low-fidelity model with a subgrid-scale model to reduce the computational cost, but selecting the appropriate subgrid-scale models and tuning them are challenging. We propose a novel method for learning the subgrid-scale model effects when simulating partial differential equations using neural ordinary differential equations in the context of discontinuous Galerkin (DG) spatial discretization. Our approach learns the missing scales of the low-order DG solver at a continuous level and hence improves the accuracy of the low-order DG approximations as well as accelerates the filtered high-order DG simulations with a certain degree of precision. We demonstrate the performance of our approach through multidimensional Taylor--Green vortex examples at different Reynolds numbers and times, which cover laminar, transitional, and turbulent regimes. The proposed method not only reconstructs the subgrid-scale from the low-order (1st-order) approximation but also speeds up the filtered high-order DG (6th-order) simulation by two orders of magnitude.

通过使用神经常微分方程在间断Galerkin空间离散化的背景下，我们提出了一种学习部分微分方程模拟时子网格规模模型效应的新方法，通过连续水平学习低阶DG求解器中丢失的尺度，从而提高低阶DG逼近的准确性并以一定程度的精度加速滤波器高阶DG模拟。我们通过不同雷诺数和时间的多维Taylor-Green涡旋实例演示了我们方法的性能，涵盖了层流、过渡和湍流状态。提出的方法不仅可以从低阶（1阶）逼近重建子网格规模，还可以将滤波高阶（6阶）DG模拟加速两个数量级。

Learning Subgrid-Scale Models in Discontinuous Galerkin Methods with
  Neural Ordinary Differential Equations for Compressible Navier--Stokes
  Equations

利用神经常微分方程在不连续Galerkin方法中学习亚网颗粒模型以处理可压性Navier-Stokes方程