Establish algebraic data-driven constitutive models for elastic solids with a tensorial sparse symbolic regression method and a hybrid feature selection technique

We propose a tensorial sparse symbolic regression method to directly establish explicit algebraic tensorial macroscopic constitutive models for elastic solids under large deformation from data obtained at the mesoscale. Unlike “black-box” data-driven models such as artificial neural networks, the model established with our proposed method is a “white-box” polynomial of tensors, which is important for gaining insights into the deformation nature from multi-scale calculations and modeling. The proposed method first generates a massive number of candidate basis tensor terms and then combines them linearly with fitting coefficients, only a small set of which are non-zero. In this way, the established constitutive models are concise and precise. An innovative hybrid feature selection and regression technique (a combination of high-dimensional Lasso, teaching–learning-based optimization and recursive feature elimination) is proposed to overcome the issues of high correlation and multicollinearity of the terms in this sparse regression problem. Four benchmark tests are introduced to verify that the proposed method established correct models directly from data generated by the benchmarks. A rate-type hypoelastic model is also established with the data generated by the not rate-type benchmarks. At last, an example of constitutive modeling for particle-reinforced composites is introduced to illustrate the ability of the proposed method to build explicit tensorial macroscopic models from data generated by mesoscopic calculations, demonstrating its potential for widespread application in multi-scale simulations of hierarchical materials.