Learning Invariance Preserving Moment Closure Model for Boltzmann–BGK Equation

As one of the main governing equations in kinetic theory, Boltzmann equation is widely utilized aerospace, microscopic flow, etc. Its high-resolution simulation crucial these related areas. However, due to high dimensionality equation, simulations are often difficult achieve numerically. The moment method which was first proposed Grad (Commun Pure Appl Math 2(4):331–407, 1949) among popular numerical methods efficient simulations. We can derive by taking moments on both sides effectively reduces problem. challenges that it leads an unclosed system, and closure needed obtain a closed system. It truly art designing closures for systems has been significant research field theory. Other than traditional human designs closures, machine learning-based approach attracted much attention lately Han et al. (Proc Natl Acad Sci USA 116(44):21983–21991, 2019) Huang (J Non-Equilib Thermodyn 46(4):355–370, 2021). In this work, we propose model Boltzmann–BGK equation. particular, relation approximated carefully designed deep neural network possesses desirable physical invariances, i.e., Galilean invariance, reflecting scaling inherited from original playing important role correct Numerical 1D–1D examples including smooth discontinuous initial condition problems, Sod shock tube problem, structure 1D–3D problems demonstrate satisfactory performances invariance preserving method.