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

Neural network-based, structure-preserving entropy closures for the Boltzmann moment system