A structure-preserving surrogate model for the closure of the moment system of the Boltzmann equation using convex deep neural networks

• Main Authors: Schotthöfer, Steffen, Xiao, Tianbai, Frank, Martin, Hauck, Cory D
• Format: Journal Article
• Language: English
• Published: 17.06.2021
• Subjects: Computer Science - Numerical Analysis; Mathematics - Mathematical Physics; Mathematics - Numerical Analysis; Physics - Mathematical Physics
• DOI: 10.48550/arxiv.2106.09445

Direct simulation of physical processes on a kinetic level is prohibitively expensive in aerospace applications due to the extremely high dimension of the solution spaces. In this paper, we consider the moment system of the Boltzmann equation, which projects the kinetic physics onto the hydrodynamic scale. The unclosed moment system can be solved in conjunction with the entropy closure strategy. Using an entropy closure provides structural benefits to the physical system of partial differential equations. Usually computing such closure of the system spends the majority of the total computational cost, since one needs to solve an ill-conditioned constrained optimization problem. Therefore, we build a neural network surrogate model to close the moment system, which preserves the structural properties of the system by design, but reduces the computational cost significantly. Numerical experiments are conducted to illustrate the performance of the current method in comparison to the traditional closure.