Natural model reduction for kinetic equations (Q6588181)

A promising strategy for tackling high-dimensional problems is to uncover their intrinsically low-dimensional structures, often achieved through modern techniques for low-dimensional function representation, such as machine learning. Building on finite-dimensional approximate solution manifolds, this paper introduces a novel model reduction framework for kinetic equations.\N\NThe proposed approach uses projections onto the tangent bundles of these approximate manifolds, leading naturally to first-order hyperbolic systems. When certain conditions are met, the reduced models retain several key properties, including hyperbolicity, conservation laws, entropy dissipation, finite propagation speed, and linear stability.\N\NThis work provides a rigorous analysis of the connection between the H-theorem for kinetic equations and the linear stability of the reduced systems, offering insights into the selection of Riemannian metrics for model reduction. The framework is versatile and applicable to the reduction of various models in kinetic theory.