  Building efficient, accurate and generalizable reduced order models of
developed turbulence remains a major challenge. This manuscript approaches this
problem by developing a hierarchy of parameterized reduced Lagrangian models
for turbulent flows, and investigates the effects of enforcing physical
structure through Smoothed Particle Hydrodynamics (SPH) versus relying on
neural networks (NN)s as universal function approximators. Starting from Neural
Network (NN) parameterizations of a Lagrangian acceleration operator, this
hierarchy of models gradually incorporates a weakly compressible and
parameterized SPH framework, which enforces physical symmetries, such as
Galilean, rotational and translational invariances. Within this hierarchy, two
new parameterized smoothing kernels are developed in order to increase the
flexibility of the learn-able SPH simulators. For each model we experiment with
different loss functions which are minimized using gradient based optimization,
where efficient computations of gradients are obtained by using Automatic
Differentiation (AD) and Sensitivity Analysis (SA). Each model within the
hierarchy is trained on two data sets associated with weekly compressible
Homogeneous Isotropic Turbulence (HIT): (1) a validation set using weakly
compressible SPH; and (2) a high fidelity set from Direct Numerical Simulations
(DNS). Numerical evidence shows that encoding more SPH structure improves
generalizability to different turbulent Mach numbers and time shifts, and that
including the novel parameterized smoothing kernels improves the accuracy of
SPH at the resolved scales.
