  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. Therefore, reliable and
accurate closure models for 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-Benard 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.
