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.

翻译：近年来计算能力的持续提升使数值模拟日趋复杂和精确。然而，高保真度模拟虽对科学发现和问题求解具有重要价值，但伴随巨大的计算需求。因此，通常采用带有亚网格尺度模型的低保真度模型来降低计算成本，但如何选择合适的亚网格尺度模型并对其进行调谐仍具挑战性。本文提出一种新颖方法，在不连续伽辽金（DG）空间离散框架下，利用神经常微分方程学习偏微分方程模拟中的亚网格尺度模型效应。该方法在连续层面学习低阶DG求解器缺失的尺度信息，从而既提升低阶DG近似的精度，又能以特定精度加速滤波高阶DG模拟。我们通过不同雷诺数和时间下的多维泰勒-格林涡算例（涵盖层流、转捩和湍流状态）验证了该方法性能。实验表明，所提方法不仅从低阶（一阶）近似中重建了亚网格尺度，还将滤波高阶DG（六阶）模拟的速度提升两个数量级。