Learning to integrate non-linear equations from highly resolved direct numerical simulations (DNSs) has seen recent interest for reducing the computational load for fluid simulations. Here, we focus on determining a flux-limiter for shock capturing methods. Focusing on flux limiters provides a specific plug-and-play component for existing numerical methods. Since their introduction, an array of flux limiters has been designed. Using the coarse-grained Burgers' equation, we show that flux-limiters may be rank-ordered in terms of their log-error relative to high-resolution data. We then develop theory to find an optimal flux-limiter and present flux-limiters that outperform others tested for integrating Burgers' equation on lattices with $2\times$, $3\times$, $4\times$, and $8\times$ coarse-grainings. We train a continuous piecewise linear limiter by minimizing the mean-squared misfit to 6-grid point segments of high-resolution data, averaged over all segments. While flux limiters are generally designed to have an output of $\phi(r) = 1$ at a flux ratio of $r = 1$, our limiters are not bound by this rule, and yet produce a smaller error than standard limiters. We find that our machine learned limiters have distinctive features that may provide new rules-of-thumb for the development of improved limiters. Additionally, we use our theory to learn flux-limiters that outperform standard limiters across a range of values (as opposed to at a specific fixed value) of coarse-graining, number of discretized bins, and diffusion parameter. This demonstrates the ability to produce flux limiters that should be more broadly useful than standard limiters for general applications.

翻译：学习从高分辨率直接数值模拟（DNS）中积分非线性方程，近年来因能降低流体模拟计算负担而受到关注。本文聚焦于为激波捕获方法确定通量限制器。关注通量限制器可为现有数值方法提供一种特定的即插即用组件。自其提出以来，已设计出多种通量限制器。利用粗粒化的伯格斯方程，我们展示了通量限制器可根据其相对于高分辨率数据的对数误差进行排序。随后，我们发展理论以寻找最优通量限制器，并提出了在$2\times$、$3\times$、$4\times$和$8\times$粗粒化网格上积分伯格斯方程时优于其他测试结果的通量限制器。通过最小化高分辨率数据中6网格点片段（对所有片段取平均）的均方误差，我们训练了一个连续分段线性限制器。尽管通量限制器通常设计为在通量比$r=1$时输出$\phi(r) = 1$，但我们的限制器不受此规则约束，却产生了比标准限制器更小的误差。我们发现，机器学习得到的限制器具有独特特征，可为开发改进型限制器提供新的经验法则。此外，我们利用理论学习了通量限制器，其在粗粒化程度、离散区间数量和扩散参数等不同取值范围（而非固定值）内均优于标准限制器。这表明，所生成的通量限制器在通用应用中应比标准限制器更具广泛适用性。