  One of the biggest challenges for simulating the Boltzmann equation is the
evaluation of fivefold collision integral. Given the recent successes of deep
learning and the availability of efficient tools, it is an obvious idea to try
to substitute the evaluation of the collision operator by the evaluation of a
neural network. However, it is unlcear whether this preserves key properties of
the Boltzmann equation, such as conservation, invariances, the H-theorem, and
fluid-dynamic limits. In this paper, we present an approach that guarantees the
conservation properties and the correct fluid dynamic limit at leading order.
The concept originates from a recently developed scientific machine learning
strategy which has been named "universal differential equations". It proposes a
hybridization that fuses the deep physical insights from classical Boltzmann
modeling and the desirable computational efficiency from neural network
surrogates. The construction of the method and the training strategy are
demonstrated in detail. We conduct an asymptotic analysis and illustrate its
multi-scale applicability. The numerical algorithm for solving the neural
network-enhanced Boltzmann equation is presented as well. Several numerical
test cases are investigated. The results of numerical experiments show that the
time-series modeling strategy enjoys the training efficiency on this supervised
learning task.
