Simulating complex physical systems often involves solving partial differential equations (PDEs) with some closures due to the presence of multi-scale physics that cannot be fully resolved. Although the advancement of high performance computing has made resolving small-scale physics possible, such simulations are still very expensive. Therefore, reliable and accurate closure models for the unresolved physics remains an important requirement for many computational physics problems, e.g., turbulence simulation. Recently, several researchers have adopted generative adversarial networks (GANs), a novel paradigm of training machine learning models, to generate solutions of PDEs-governed complex systems without having to numerically solve these PDEs. However, GANs are known to be difficult in training and likely to converge to local minima, where the generated samples do not capture the true statistics of the training data. In this work, we present a statistical constrained generative adversarial network by enforcing constraints of covariance from the training data, which results in an improved machine-learning-based emulator to capture the statistics of the training data generated by solving fully resolved PDEs. We show that such a statistical regularization leads to better performance compared to standard GANs, measured by (1) the constrained model's ability to more faithfully emulate certain physical properties of the system and (2) the significantly reduced (by up to 80%) training time to reach the solution. We exemplify this approach on the Rayleigh-Bénard convection, a turbulent flow system that is an idealized model of the Earth's atmosphere. With the growth of high-fidelity simulation databases of physical systems, this work suggests great potential for being an alternative to the explicit modeling of closures or parameterizations for unresolved physics, which are known to be a major source of uncertainty in simulating multi-scale physical systems, e.g., turbulence or Earth's climate.

由于存在无法完全解决的多尺度物理问题，因此模拟复杂的物理系统通常涉及求解带有某些闭包的偏微分方程（PDE）。尽管高性能计算的进步已经使解决小规模的物理问题成为可能，但这种模拟仍然非常昂贵。因此，对于尚未解决的物理问题，可靠而准确的闭合模型仍然是许多计算物理问题（例如湍流模拟）的重要要求。最近，一些研究人员采用了生成对抗性网络（GANs）（一种训练机器学习模型的新颖范例）来生成PDE管辖的复杂系统的解决方案，而不必对这些PDE进行数值求解。但是，众所周知，GAN很难训练，并且可能会收敛到局部最小值，其中生成的样本未捕获训练数据的真实统计信息。在这项工作中，我们通过从训练数据中强制协方差约束，提出了一个统计约束的生成对抗网络，这导致了一种改进的基于机器学习的仿真器，可以捕获通过求解完全解析的PDE生成的训练数据的统计信息。我们表明，与标准GAN相比，这样的统计正则化导致更好的性能，可以通过以下方法衡量：（1）约束模型更忠实地模拟系统的某些物理属性的能力，以及（2）显着减少（最多80％）的训练是时候找到解决办法了。我们以瑞利-贝纳德对流（Rayleigh-Bénard对流）为例，这种对流是一种湍流系统，是地球大气的理想模型。