  High Reynolds Homogeneous Isotropic Turbulence is fully described within the
Navier-Stokes (NS) equations, which are notoriously difficult to solve
numerically. Engineers, interested primarily in describing turbulence at a
reduced range of resolved scales, have designed heuristics, known as Large Eddy
Simulation (LES). LES is described in terms of the temporally evolving Eulerian
velocity field defined over a spatial grid with the mean-spacing correspondent
to the resolved scale. This classic Eulerian LES depends on assumptions about
effects of sub-grid scales on the resolved scales.
  Here, we take an alternative approach and design novel LES heuristics stated
in terms of Lagrangian particles moving with the flow. Our Lagrangian LES, thus
L-LES, is described by equations generalizing the weakly compressible Smoothed
Particle Hydrodynamics formulation with extended parametric and functional
freedom, which is then resolved via Machine Learning training on Lagrangian
data from Direct Numerical Simulations of the NS equations. The L-LES model
includes physics-informed parameterization and functional form, by combining
physics-based parameters and physics-inspired Neural Networks 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. We
experiment with loss functions of different types, including physics-informed
ones accounting for statistics of Lagrangian particles. 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.
