Dynamical Low-Rank Matrix and Tensor-Network Methods in Fluid Mechanics

Abstract

Fluid flows exhibit correlated structures across space and time. Dynamical low-rank approximation begins by representing the state variable as a matrix or tensor, extracting and exploiting its evolving correlated structure on the fly to construct efficient reduced-order models directly from the governing equations. Tensor-network constructions, originally developed in quantum many-body physics, extend this framework to high-dimensional problems by providing efficient representations that mitigate the curse of dimensionality. These methods have found applications across fluid mechanics, including flow stability analysis, uncertainty quantification, turbulent combustion, and quantum-inspired formulations based on tensor quantics. Complementing these developments, cross-interpolation techniques provide scalable algorithms that compute and evolve only a targeted reduced set of dynamical degrees of freedom, avoiding the construction of full matrices and tensors. This review surveys the mathematical foundations, algorithms, and applications of dynamical low-rank approximation, tensor networks, and cross interpolation in fluid mechanics.