Adaptive and hybrid reduced order models to mitigate Kolmogorov barrier in a multiscale kinetic transport equation

Abstract: In this work, we develop reduced order models (ROMs) to predict solutions to a multiscale kinetic transport equation with a diffusion limit under the parametric setting. When the underlying scattering effect is not sufficiently strong, the system governed by this equation exhibits transport-dominated behavior. Suffering from the Kolmogorov barrier for transport-dominant problems, classical linear ROMs may become inefficient in this regime. To address this issue, we first develop a piecewise linear ROM by introducing a novel goal-oriented adaptive time partitioning strategy. To avoid local over-refinement or under-refinement, we propose an adaptive coarsening and refinement strategy that remains robust with various initial empirical partitions. Additionally, for problems where a local linear approximation is not sufficiently efficient, we further develop a hybrid ROM, which combines autoencoder-based nonlinear ROMs and piecewise linear ROMs. Compared to previous autoencoder-based ROMs, this hybridized method reduces the offline autoencoder's training cost by only applying it to time intervals that are adaptively identified as the most challenging. Numerical experiments demonstrate that our proposed approaches successfully predict full-order solutions at unseen parameter values with both efficiency and accuracy. To the best of our knowledge, this is the first attempt to address the Kolmogorov barrier for multiscale kinetic transport problems with the coexistence of both transport- and diffusion-dominant behaviors.