Regression-based projection for learning Mori-Zwanzig operators for isotropic turbulence

The Mori-Zwanzig (MZ) formalism provides a mathematically rigorous procedure for constructing reduced-order representations of high-dimensional dynamical systems, where the effect due to the unresolved dynamics are captured in the memory kernel and orthogonal dynamics. Our previous work on data-driven learning of MZ operators based on Mori's projection operator demonstrated successful extraction of these operators for homogeneous isotropic turbulence. However, the linearity of Mori's projection operator significantly limits the applicability of the algorithm to turbulence modeling. To bridge the gap between Mori's linear and Zwanzig's projection operators, our group developed a more general MZ learning algorithm that adopts statistical regression as a projection operator. We experiment with a range of regression-based models, including those connected with existing turbulence modeling frameworks, for learning non-Markovian models based on MZ formalism from DNS datasets. We show that the extracted operators exhibit improved performance compared to previous linear projection-based MZ operators, depending on the complexity of the regression models. The regression-based MZ learning algorithm provides a promising framework for developing turbulence models with memory effects.