  Reduced models describing the Lagrangian dynamics of the Velocity Gradient
Tensor (VGT) in Homogeneous Isotropic Turbulence (HIT) are developed under the
Physics-Informed Machine Learning (PIML) framework. We consider VGT at both
Kolmogorov scale and coarse-grained scale within the inertial range of HIT.
Building reduced models requires resolving the pressure Hessian and sub-filter
contributions, which is accomplished by constructing them using the integrity
bases and invariants of VGT. The developed models can be expressed using the
extended Tensor Basis Neural Network (TBNN). Physical constraints, such as
Galilean invariance, rotational invariance, and incompressibility condition,
are thus embedded in the models explicitly. Our PIML models are trained on the
Lagrangian data from a high-Reynolds number Direct Numerical Simulation (DNS).
To validate the results, we perform a comprehensive out-of-sample test. We
observe that the PIML model provides an improved representation for the
magnitude and orientation of the small-scale pressure Hessian contributions.
Statistics of the flow, as indicated by the joint PDF of second and third
invariants of the VGT, show good agreement with the "ground-truth" DNS data. A
number of other important features describing the structure of HIT are
reproduced by the model successfully. We have also identified challenges in
modeling inertial range dynamics, which indicates that a richer modeling
strategy is required. This helps us identify important directions for future
research, in particular towards including inertial range geometry into TBNN.
