  We develop a learning strategy to infer the constitutive relation for the
stress of polymeric flows with memory. We make no assumptions regarding the
functional form of the constitutive relations, except that they should be
expressible in differential form as a function of the local stress- and
strain-rate tensors. In particular, we use a Gaussian Process regression to
infer the constitutive relations from stress trajectories generated from
small-scale (fixed strain-rate) microscopic polymer simulations. For
simplicity, a Hookean dumbbell representation is used as a microscopic model,
but the method itself can be generalized to incorporate more realistic
descriptions. The learned constitutive relation is then used to perform
macroscopic flow simulations, allowing us to update the stress distribution in
the fluid in a manner that accounts for the microscopic polymer dynamics. The
results using the learned constitutive relation are in excellent agreement with
full Multi-Scale Simulations, which directly couple micro/macro degrees of
freedom, as well as the exact analytical solution given by the Maxwell
constitutive relation. We are able to fully capture the history dependence of
the flow, as well as the elastic effects in the fluid. We expect the proposed
learning/simulation approach to be used not only to study the dynamics of
entangled polymer flows, but also for the complex dynamics of other Soft Matter
systems, which possess a similar hierarchy of length- and time-scales.
