Machine Learning Statistical Evolution of the Coarse-Grained Velocity Gradient Tensor

We exploit recent advances in physics-informed machine learning and phenomenological theories of turbulence to develop parameterized stochastic differential equations (SDEs) coupling the Lagrangian evolution of a fluid volume to the coarse-grained velocity gradient tensor; resulting in a reduced order model for incompressible turbulence. Choosing minimal representations of fluid geometry and velocity gradient tensor, we search for local approximations to nonlinear (pressure and subgrid) terms. The goal is achieved by optimizing physics-informed neural networks - dependent on the coupled system of fluid geometry and velocity gradient tensor - over high fidelity Lagrangian direct numerical simulation data. We demonstrate the ability of the parameterized SDEs to reproduce the topological statistics of the coarse-grained velocity gradient tensor, as well as the shape distributions of the respective fluid elements.