Projection-based model reduction for thermo-chemical non-equilibrium gas mixtures*

State-specific thermochemical collisional models are crucial for accurately describing the physics of systems involving non-equilibrium plasmas. Although these (nonlinear) models provide detailed insights into kinetic processes when internal energy levels significantly deviate from the equilibrium Maxwell-Boltzmann distribution, they are computationally expensive and impractical for large-scale, multi-dimensional simulations. While computational cost can be reduced significantly by using low-order models, these are typically derived using empirical arguments and physics-based assumptions that ultimately lead to inaccurate predictions. We propose to address this issue by developing projection-based reduced-order models that leverage the form of full-order governing equations. In particular, we identify an oblique projection operator by balancing the state and gradient covariance matrices associated with the linearization of the full-order equations about thermochemical-equilibrium steady-state solutions. The use of the linearized equations allows for fast data generation and training, while also providing a good approximation of the low-order subspace the nonlinear dynamics evolve on. Once the projection operator is identified, we obtain a reduced-order model via Petrov-Galerkin projection of the original nonlinear system. The overall procedure is akin to the well-known balanced truncation for linear time-invariant systems, and to the recently-developed CoBRAS formulation proposed by Otto et al., SISC, 2023. Reduced-order models are developed and tested on two distinct thermochemical systems: i) a  rovibrational collisional model for the O₂-O system, and ii) a vibrational collisional model for the combined O₂-O and O₂-O₂ systems. Our approach demonstrates both high accuracy and significant computational speedup for these nonlinear chemical kinetic systems, with training times on the order of minutes on a single GPU. The model effectively captures differences in orders of magnitude within the quantum state distribution function, resolving both low and high lying energy states.