Tensor networks enable the calculation of turbulence probability distributions

Abstract: Predicting the dynamics of turbulent fluid flows has long been a central goal of science and engineering. Yet, even with modern computing technology, accurate simulation of all but the simplest turbulent flow-fields remains impossible: the fields are too chaotic and multi-scaled to directly store them in memory and perform time-evolution. An alternative is to treat turbulence $\textit{probabilistically}$, viewing flow properties as random variables distributed according to joint probability density functions (PDFs). Turbulence PDFs are neither chaotic nor multi-scale, but are still challenging to simulate due to their high dimensionality. Here we show how to overcome the dimensionality problem by parameterising turbulence PDFs into an extremely compressed format known as a "tensor network" (TN). The TN paradigm enables simulations on single CPU cores 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 $\mathcal{O}(10^6)$ and $\mathcal{O}(10^3)$, respectively, compared to standard finite difference algorithms. A future path is opened towards something heretofore regarded as infeasible: directly simulating high-dimensional PDFs of both turbulent flows and other chaotic systems that are useful to describe probabilistically.