A neural network architecture and training algorithm to predict viscoelastic stresses from vortical data

Abstract

Numerical simulations of elastic turbulence in parallel shear flows of polymer solutions indicate that the phenomena is associated with the formation and instability of exact coherent states dominated by thin sheets of polymer stress. However, these ``arrowhead'' structures are yet to be seen directly in experiments, where simultaneous velocity and polymer conformation measurements are challenging to obtain. Motivated by these challenges, we introduce a method for the prediction of the polymer conformation field given a time series of vorticity measurements. Our approach consists of two components: the first is a convolutional neural network architecture which takes vorticity fields and outputs a positive definite conformation tensor. The second is the adaptation of an assimilation-based training algorithm (Zhu \& Page, 2026) which does not require a pre-generated `offline' library of reference conformation fields, but is trained only using the vorticity measurements. This is particularly important in viscoelastic problems, where the appropriate model and parameters to compare to the experiments may need to be determined as part of the solution. In training, measurements made on a time-marched network prediction are required to match the saved time series, while the output of the solver and network predictions at later times are required to be self-consistent. We apply these ideas to two-dimensional Kolmogorov flow in a range of regimes, from simple traveling waves to a fully chaotic state. In all cases, our method produces robust predictions of the polymer stretch, while standard, unregularised variational assimilation is ineffective. In the chaotic case we show that our networks generalise to much larger domains -- without further optimisation -- than the `minimal' units in which they were trained.