Sparse Regression for Discovery of Constitutive Models from Oscillatory Shear Measurements

Abstract

We propose sparse regression as an alternative to neural networks for the discovery of parsimonious constitutive models (CMs) from oscillatory shear experiments. Symmetry and frame-invariance are strictly imposed by using tensor basis functions to isolate and describe unknown nonlinear terms in the CMs. We generate synthetic experimental data using the Giesekus and Phan-Thien Tanner CMs, and consider two different scenarios. In the complete information scenario, we assume that the shear stress, along with the first and second normal stress differences, is measured. This leads to a sparse linear regression problem that can be solved efficiently using $l_1$ regularization. In the partial information scenario, we assume that only shear stress data is available. This leads to a more challenging sparse nonlinear regression problem, for which we propose a greedy two-stage algorithm. In both scenarios, the proposed methods fit and interpolate the training data remarkably well. Predictions of the inferred CMs extrapolate satisfactorily beyond the range of training data for oscillatory shear. They also extrapolate reasonably well to flow conditions like startup of steady and uniaxial extension that are not used in the identification of CMs. We discuss ramifications for experimental design, potential algorithmic improvements, and implications of the non-uniqueness of CMs inferred from partial information.