Solving fluid dynamics equations often requires the use of closure relations
that account for missing microphysics. For example, when solving equations
related to fluid dynamics for systems with a large Reynolds number, sub-grid
effects become important and a turbulence closure is required, and in systems
with a large Knudsen number, kinetic effects become important and a kinetic
closure is required. By adding an equation governing the growth and transport
of the quantity requiring the closure relation, it becomes possible to capture
microphysics through the introduction of ``hidden variables'' that are
non-local in space and time. The behavior of the ``hidden variables'' in
response to the fluid conditions can be learned from a higher fidelity or
ab-initio model that contains all the microphysics. In our study, a partial
differential equation simulator that is end-to-end differentiable is used to
train judiciously placed neural networks against ground-truth simulations. We
show that this method enables an Euler equation based approach to reproduce
non-linear, large Knudsen number plasma physics that can otherwise only be
modeled using Boltzmann-like equation simulators such as Vlasov or
Particle-In-Cell modeling.

通过添加引入 “隐藏变量” 的传输及生长方程，使用可完全微分的偏微分方程模拟器训练神经网络的方法，实现了基于 Euler 方程的流体动力学建模，可以对大 Knudsen 数等非线性等离子体物理现象进行建模。

多尺度流体模拟中隐含变量的机器学习

Machine learning of hidden variables in multiscale fluid simulation

In this investigation, a data-driven turbulence closure framework is
introduced and deployed for the sub-grid modelling of Kraichnan turbulence. The
novelty of the proposed method lies in the fact that snapshots from
high-fidelity numerical data are used to inform artificial neural networks for
predicting the turbulence source term through localized grid-resolved
information. In particular, our proposed methodology successfully establishes a
map between inputs given by stencils of the vorticity and the streamfunction
along with information from two well-known eddy-viscosity kernels. Through this
we predict the sub-grid vorticity forcing in a temporally and spatially dynamic
fashion. Our study is both a-priori and a-posteriori in nature. In the former,
we present an extensive hyper-parameter optimization analysis in addition to
learning quantification through probability density function based validation
of sub-grid predictions. In the latter, we analyse the performance of our
framework for flow evolution in a classical decaying two-dimensional turbulence
test case in the presence of errors related to temporal and spatial
discretization. Statistical assessments in the form of angle-averaged kinetic
energy spectra demonstrate the promise of the proposed methodology for sub-grid
quantity inference. In addition, it is also observed that some measure of
a-posteriori error must be considered during optimal model selection for
greater accuracy. The results in this article thus represent a promising
development in the formalization of a framework for generation of
heuristic-free turbulence closures from data.

本文提出一种数据驱动的湍流封闭框架，利用人工神经网络预测湍流源项，建立了多个输入和输出之间的映射关系，优化分析和概率密度函数验证了子网格预测，证明了该方法可推断子网格量，为实现无启发式湍流封闭算法打下了良好的基础。