  We introduce a machine-learning-based framework for constructing continuum
non-Newtonian fluid dynamics model directly from a micro-scale description.
Dumbbell polymer solutions are used as examples to demonstrate the essential
ideas. To faithfully retain molecular fidelity, we establish a micro-macro
correspondence via a set of encoders for the micro-scale polymer configurations
and their macro-scale counterparts, a set of nonlinear conformation tensors.
The dynamics of these conformation tensors can be derived from the micro-scale
model and the relevant terms can be parametrized using machine learning. The
final model named the deep non-Newtonian model (DeePN$^2$), takes the form of
conventional non-Newtonian fluid dynamics models, with a new form of the
objective tensor derivative. Both the formulation of the dynamic equation and
the neural network representation rigorously preserve the rotational
invariance, which ensures the admissibility of the constructed model. Numerical
results demonstrate the accuracy of DeePN$^2$, where models based on empirical
closures show limitations.

Computational Physics;Numerical Analysis;Numerical Analysis;Fluid Dynamics;Machine Learning