Structure-preserving machine learning moment closures for kinetic equations

Kinetic theory describes the behaviors of dynamic systems from a statistical point of view. It has wide applications in many fields, including supersonic flows, microelectromechanical systems, unconventional gas reservoirs, space vehicle re-entry problems, and nuclear fusion. Because of the high dimensionality of such models, efficient simulation is a long-standing challenge, which limits their applications to real-world problems. This research project will address this challenge by developing reduced models to approximate the kinetic equations. These models, called moment models, are expected to capture the physics and serve as good surrogates with the aid of machine learning (ML). This will provide a powerful tool in the modeling and simulation of non-equilibrium phenomena in physics and engineering. The project will provide research opportunities for graduate and undergraduate students who are interested in computational mathematics, and provide curriculum development in the PI's department. The primary objective of this research is to develop robust, accurate, and efficient ML moment models with some provable mathematical structures. The project focuses on how to preserve the hyperbolicity structure of the ML moment models. The hyperbolicity is closely related to the well-posedness of the first-order system of partial differential equations and is also vitally important for robust numerical simulations. The following ideas and methodologies will be investigated: (1) a symmetrizer-based approach and an eigenvalue-based approach that preserve the hyperbolicity of the model in multidimensional cases by exploiting the algebraic structure of the ML moment model; (2) a ML approach to learning boundary conditions that ensures necessary conditions for the well-posedness of the initial boundary value problem for the moment model; (3) a ML model with hyperbolicity enforced by generalized data-driven moments. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.