Lagrangian Large Eddy Simulations via Physics-Informed Machine Learning*

In this work, we apply Physics-Informed Machine Learning to develop Lagrangian Large Eddy Simulation (L-LES) models for turbulent flows. We generalize the evolutionary equations of Lagrangian particles moving in weakly compressible turbulence with extended, physics-informed parametrization and functional freedom, by combining physics-based parameters and physics-inspired Neural Networks (NN) to describe the evolution of turbulence within the resolved range of scales. The sub-grid scale contributions are modeled separately with physical constraints to account for the effects from un-resolved scales. We build the resulting model under the Differentiable Programming framework to facilitate efficient training and then train the model on a set of coarse-grained Lagrangian data extracted from fully-resolved Direct Numerical Simulations. We experiment with loss functions of different types, including trajectory, field, and statistics-based ones to embed physics into the learning. We show that our Lagrangian LES model is capable of reproducing Eulerian and unique Lagrangian turbulence structures and statistics over a range of turbulent Mach numbers.