  A promising approach to investigating high-dimensional problems is to
identify their intrinsically low-dimensional features, which can be achieved
through recently developed techniques for effective low-dimensional
representation of functions such as machine learning. Based on available
finite-dimensional approximate solution manifolds, this paper proposes a novel
model reduction framework for kinetic equations. The method employs projections
onto tangent bundles of approximate manifolds, naturally resulting in
first-order hyperbolic systems. Under certain conditions on the approximate
manifolds, the reduced models preserve several crucial properties, including
hyperbolicity, conservation laws, entropy dissipation, finite propagation
speed, and linear stability. For the first time, this paper rigorously
discusses the relation between the H-theorem of kinetic equations and the
linear stability conditions of reduced systems, determining the choice of
Riemannian metrics involved in the model reduction. The framework is widely
applicable for the model reduction of many models in kinetic theory.
