Tensor networks enable the calculation of turbulence probability distributions

Predicting the dynamics of turbulent fluids has been an elusive goal for centuries. Even with modern computers, anything beyond the simplest turbulent flows is too chaotic and multiscaled to be directly simulatable. An alternative is to treat turbulence probabilistically, viewing flow properties as random variables distributed according to joint probability density functions (PDFs). Such PDFs are neither chaotic nor multiscale, yet remain challenging to simulate due to their high dimensionality. Here, we overcome the dimensionality problem by encoding turbulence PDFs as highly compressed "tensor networks" (TNs). This enables single CPU core simulations that would otherwise be impractical even with supercomputers: for a 5 + 1 dimensional PDF of a chemically reactive turbulent flow, we achieve reductions in memory and computational costs by factors of [Formula: see text] and [Formula: see text], respectively, compared to standard finite-difference algorithms. A future path is opened toward something heretofore thought infeasible: directly simulating high-dimensional PDFs of both turbulent flows and other chaotic systems that can usefully be described probabilistically.