The growing computing power over the years has enabled simulations to become more complex and accurate. While immensely valuable for scientific discovery and problem-solving, however, high-fidelity simulations 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 augmented by neural ordinary differential operators 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：随着计算能力的持续增长，数值模拟的复杂性和精确性不断提升。虽然高保真模拟对科学发现和问题求解具有巨大价值，但其计算成本极高。因此，通常采用配备亚网格尺度模型的低保真模型来降低计算成本，然而选择恰当的亚网格模型并进行参数调整颇具挑战性。本文提出一种新颖方法，在间断伽辽金（DG）空间离散框架下，通过神经常微分算子增强偏微分方程模拟，学习亚网格尺度模型效应。该方法在连续层面上捕捉低阶DG求解器缺失的尺度信息，从而提升低阶DG近似的精度，并在保持一定精度的前提下加速滤波高阶DG模拟。通过不同雷诺数和时间参数下的多维泰勒-格林涡旋算例（涵盖层流、过渡流和湍流区域）验证了算法的性能。实验表明，本文方法不仅能从低阶（一阶）近似中重建亚网格尺度，还可将滤波高阶DG（六阶）模拟速度提升两个数量级。