Compression, simulation, and synthesis of turbulent flows with tensor trains

Abstract

Numerical simulations of turbulent fluids are paramount to real-life applications, from predicting and modeling flows to diagnostic purposes in engineering. However, they are also computationally challenging due to their intrinsically non-linear dynamics, which require a very high spatial resolution to accurately describe them. A promising idea is to represent flows on a discrete mesh using tensor trains (TTs), featuring a convenient scaling of the number of parameters with the mesh size. However, it is unclear how the compression power of TTs is affected by the complexity of the flows, as measured by the Reynolds number. In fact, no comprehensive analysis of how the TT representation affects the turbulent properties has yet been carried out. We fill this gap by analyzing TTs as an Ansatz to compress, simulate, and generate 3D snapshots with turbulent-like features. Specifically, we first investigate the effect of TT compression on key turbulence signatures, such as the energy spectrum, the PDF of velocity increments, and flatness. Second, we extend the 2D TT-solver introduced in [1] to a 3D cubic domain with periodic boundary conditions. We use it to simulate the incompressible Navier-Stokes dynamics at $Re_{\lambda}=315$ for a total of 9-10 Kolmogorov turnover times, showcasing the numerical stability of the TT-solver in fully developed turbulent regimes. Third, we develop a TT algorithm to synthesize artificial snapshots that exhibit turbulent-like features, with a logarithmic cost in the mesh size. Our analysis demonstrates the ability of the TT representation to capture the characteristic features of turbulence. This offers a powerful quantum-inspired toolkit for the computational treatment of turbulent flows.