  This work presents neural network based minimal entropy closures for the
moment system of the Boltzmann equation, that preserve the inherent structure
of the system of partial differential equations, such as entropy dissipation
and hyperbolicity. The described method embeds convexity of the moment to
entropy map in the neural network approximation to preserve the structure of
the minimal entropy closure. Two techniques are used to implement the methods.
The first approach approximates the map between moments and the minimal entropy
of the moment system and is convex by design. The second approach approximates
the map between moments and Lagrange multipliers of the dual of the minimal
entropy optimization problem, which present the gradients of the entropy with
respect to the moments, and is enforced to be monotonic by introduction of a
penalty function. We derive an error bound for the generalization gap of convex
neural networks which are trained in Sobolev norm and use the results to
construct data sampling methods for neural network training. Numerical
experiments are conducted, which show that neural network-based entropy
closures provide a significant speedup for kinetic solvers while maintaining a
sufficient level of accuracy. The code for the described implementations can be
found in the Github repositories.
