  Important classes of active matter systems can be modeled using kinetic
theories. However, kinetic theories can be high dimensional and challenging to
simulate. Reduced-order representations based on tracking only low-order
moments of the kinetic model serve as an efficient alternative, but typically
require closure assumptions to model unrepresented higher-order moments. In
this study, we present a learning framework based on neural networks that
exploit rotational symmetries in the closure terms to learn accurate closure
models directly from kinetic simulations. The data-driven closures demonstrate
excellent a-priori predictions comparable to the state-of-the-art Bingham
closure. We provide a systematic comparison between different neural network
architectures and demonstrate that nonlocal effects can be safely ignored to
model the closure terms. We develop an active learning strategy that enables
accurate prediction of the closure terms across the entire parameter space
using a single neural network without the need for retraining. We also propose
a data-efficient training procedure based on time-stepping constraints and a
differentiable pseudo-spectral solver, which enables the learning of stable
closures suitable for a-posteriori inference. The coarse-grained simulations
equipped with data-driven closure models faithfully reproduce the mean velocity
statistics, scalar order parameters, and velocity power spectra observed in
simulations of the kinetic theory. Our differentiable framework also
facilitates the estimation of parameters in coarse-grained descriptions
conditioned on data.
