Learning Long-Term Stable Operator Inference Reduced-Order Models of Fluid Flows through Online Spatial Filtering

Abstract

This paper introduces an online evolve--filter--relax (EFR) strategy for long-term stability of Operator Inference (OpInf) reduced-order models (ROMs) of complex fluid flow simulations. The main novelty of the new EFR-OpInf strategy is the use of online (i.e., at the learned ROM online evaluation level) spatial filtering inspired from large eddy simulation to significantly improve long-term stability and predictive performance of standard OpInf. Furthermore, the EFR-OpInf strategy reduces, and in some cases even eliminates, the need for standard $L^2$ regularization, while providing physical interpretability for the OpInf hyperparameters. The EFR-OpInf framework is modular, readily integrated into existing OpInf workflows, and accommodates a user-selected ROM filtering strategy. We demonstrate the EFR-OpInf's effectiveness using a fully non-intrusive projection-based ROM filter, a ROM differential filter, and a hybrid projection-differential ROM filter. The new EFR-OpInf models are evaluated on a high-P\'eclet-number convection--diffusion--reaction problem that embeds the variation in one parameter and two unsteady Navier--Stokes problems that focus on predictions beyond a training horizon: a transitional two-dimensional flow past a cylinder and a three-dimensional turbulent minimal channel flow. Across these three scenarios, EFR-OpInf can reduce prediction errors by up to an order of magnitude relative to standard OpInf. Moreover, EFR-OpInf remains stable over long prediction horizons in cases where standard OpInf diverges. Depending on the ROM filter used, EFR-OpInf's computational cost is comparable to that of standard OpInf.