Data-driven observable discovery for reduced-order modeling of turbulence based on the Mori-Zwanzig formalism

Full-resolution simulation of turbulent flow is often computationally prohibitive, necessitating the development of reduced-order models (ROM) in real-world scientific and engineering applications. However, accurate ROM for turbulent flow is challenging, as the unresolved information of the full system influences the flow substantially. The Mori-Zwanzig (MZ) formalism provides a strategy to approach ROMs through a mathematically exact evolution of a reduced-order set of observables, in which the effects of the unresolved dynamics are captured via memory kernels and orthogonal dynamics. In our previous work [Tian et al. PoF 33(12), 2021], we presented a data-driven framework that extracts MZ kernels from Direct Numerical Simulation data, where we highlight the importance of observable choices. In this work, we aim to identify observables that can improve the learning and predictability of the learned MZ-based turbulence models. To accomplish this, we formulate a joint-learning problem by combining the learning of MZ operators with the discovery of observables from a diverse set of physics-inspired governing equations and neural network-based functions. Results show that selecting a more suitable set of observables can significantly enhance the MZ-based turbulence model's ability to predict turbulence structures and statistics at the resolved-scale.