  Reduced-order models have been widely adopted in fluid mechanics,
particularly in the context of Newtonian fluid flows. These models offer the
ability to predict complex dynamics, such as instabilities and oscillations, at
a considerably reduced computational cost.
  In contrast, the reduced-order modeling of non-Newtonian viscoelastic fluid
flows remains relatively unexplored.
  This work leverages the sparse identification of nonlinear dynamics algorithm
to develop interpretable reduced-order models for viscoelastic flows.
  In particular, we explore a benchmark oscillatory viscoelastic flow on the
four-roll mill geometry using the classical Oldroyd-B fluid.
  This flow exemplifies many canonical challenges associated with non-Newtonian
flows, including transitions, asymmetries, instabilities, and bifurcations
arising from the interplay of viscous and elastic forces, all of which require
expensive computations in order to resolve the fast timescales and long
transients characteristic of such flows.
  First, we demonstrate the effectiveness of our data-driven surrogate model to
predict the transient evolution and accurately reconstruct the spatial flow
field for fixed flow parameters. We then develop a fully parametric, nonlinear
model capable of capturing the dynamic variations as a function of the
Weissenberg number. While the training data is predominantly concentrated on a
limit cycle regime for moderate Wi, we show that the parameterized model can be
used to extrapolate, accurately predicting the dominant dynamics in the case of
high Weissenberg numbers.
  The proposed methodology represents an initial step in the field of
reduced-order modeling for viscoelastic flows with the potential to be further
refined and enhanced for the design, optimization, and control of a wide range
of non-Newtonian fluid flows using machine learning and reduced-order modeling
techniques.
