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.

翻译：本文提出采用稀疏回归替代神经网络，从振荡剪切实验中发掘简约本构模型。通过引入张量基函数严格保证对称性与框架不变性，以分离并描述本构模型中的未知非线性项。研究采用Giesekus和Phan-Thien Tanner本构模型生成合成实验数据，并考察两种不同场景：在完整信息场景中，假设剪切应力及第一、第二法向应力差均可测量，由此形成可通过$l_1$正则化高效求解的稀疏线性回归问题；在部分信息场景中，假设仅能获取剪切应力数据，此时将转化为更具挑战性的稀疏非线性回归问题，为此我们提出一种贪婪两阶段算法。两种场景下所提方法均能出色地拟合与插值训练数据。推导所得本构模型不仅能对振荡剪切训练数据范围外区域实现令人满意的预测外推，对于启动稳态拉伸与单轴拉伸等未参与模型识别的流动工况也展现出良好的外推能力。本文进一步探讨了实验设计的影响、潜在算法改进方案，以及部分信息场景下本构模型非唯一性带来的理论启示。