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：从高分辨率的直接数值模拟中学习如何积分非线性方程，近期引起了减少流体模拟计算负荷的兴趣。本文重点关注确定用于激波捕获方法的通量限制器。专注于通量限制器为现有数值方法提供了一个特定的即插即用组件。自引入以来，已设计出一系列通量限制器。利用粗粒化的Burgers方程，我们展示了通量限制器可以根据其相对于高分辨率数据的对数误差进行排序。随后，我们发展了理论以找到最优通量限制器，并提出了在$2\times$、$3\times$、$4\times$和$8\times$粗粒化网格上积分Burgers方程时优于其他测试通量限制器的方案。通过最小化所有6网格点高分辨率数据段上的均方失配，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$\phi(r) = 1$，但我们提出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，可能为改进限制器的开发提供新经验法则。此外，我们利用理论学习了在粗粒化程度、离散网格数量及扩散参数等多个值（而非固定某个值）上优于标准限制器的通量限制器，证明了通量限制器能够比标准限制器在更广泛的应用场景中发挥作用。