Tailor-Designed Models for the Turbulent Velocity Gradient through Normalizing Flow

Small-scale turbulence can be comprehensively described in terms of velocity gradients, which makes them an appealing starting point for low-dimensional modeling. Typical models consist of stochastic equations based on closures for nonlocal pressure and viscous contributions. The fidelity of the resulting models depends on the accuracy of the underlying modeling assumptions. Here, we discuss an alternative data-driven approach leveraging machine learning to derive a velocity gradient model which captures its statistics by construction. We use a normalizing flow to learn the velocity gradient probability density function (PDF) from direct numerical simulation (DNS) of incompressible turbulence. Then, by using the equation for the single-time PDF of the velocity gradient, we construct a deterministic, yet chaotic, dynamical system featuring the learned steady-state PDF by design. Finally, utilizing gauge terms for the velocity gradient single-time statistics, we optimize the time correlations as obtained from our model against the DNS data. As a result, the model time realizations resemble the time series from DNS statistically closely.