  As one of the main governing equations in kinetic theory, the Boltzmann
equation is widely utilized in aerospace, microscopic flow, etc. Its
high-resolution simulation is crucial in these related areas. However, due to
the high dimensionality of the Boltzmann equation, high-resolution simulations
are often difficult to achieve numerically. The moment method which was first
proposed by Grad is among the popular numerical methods to achieve efficient
high-resolution simulations. We can derive the governing equations in the
moment method by taking moments on both sides of the Boltzmann equation, which
effectively reduces the dimensionality of the problem. However, one of the main
challenges is that it leads to an unclosed moment system, and closure is needed
to obtain a closed moment system. It is truly an art in designing closures for
moment systems and has been a significant research field in kinetic theory.
Other than the traditional human designs of closures, the machine
learning-based approach has attracted much attention lately in Han et al. and
Huang et al. In this work, we propose a machine learning-based method to derive
a moment closure model for the Boltzmann-BGK equation. In particular, the
closure relation is approximated by a carefully designed deep neural network
that possesses desirable physical invariances, i.e., the Galilean invariance,
reflecting invariance, and scaling invariance, inherited from the original
Boltzmann-BGK equation and playing an important role in the correct simulation
of the Boltzmann equation. Numerical simulations on the 1D-1D examples
including the smooth and discontinuous initial condition problems, Sod shock
tube problem, the shock structure problems, and the 1D-3D examples including
the smooth and discontinuous problems demonstrate satisfactory numerical
performances of the proposed invariance preserving neural closure method.
