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A promising approach to investigating high-dimensional problems is toidentify their intrinsically low-dimensional features, which can be achievedthrough recently developed techniques for effective low-dimensionalrepresentation of functions such as machine learning. Based on availablefinite-dimensional approximate solution manifolds, this paper proposes a novelmodel reduction framework for kinetic equations. The method employs projectionsonto tangent bundles of approximate manifolds, naturally resulting infirst-order hyperbolic systems. Under certain conditions on the approximatemanifolds, the reduced models preserve several crucial properties, includinghyperbolicity, conservation laws, entropy dissipation, finite propagationspeed, and linear stability. For the first time, this paper rigorouslydiscusses the relation between the H-theorem of kinetic equations and thelinear stability conditions of reduced systems, determining the choice ofRiemannian metrics involved in the model reduction. The framework is widelyapplicable for the model reduction of many models in kinetic theory.

Natural Model Reduction for Kinetic Equations