Rheo-SINDy: Finding constitutive equations from nonlinear rheological data by sparse identification of nonlinear dynamics

Constitutive equations (CEs), which relate deformation rate and stress, are essential in representing macroscopic transport phenomena of complex fluids. Although finding CEs is a central subject in rheology, current methods to obtain them still need a systematic strategy; it is challenging to obtain CEs theoretically from microscopic models, and thus, macroscopic flow predictions often rely on phenomenological CEs to reproduce experimental data. This study proposes a data-driven method named Rheo-SINDy by adapting sparse identification of nonlinear dynamics (SINDy) for finding CEs from nonlinear shear rheological data. We first reconstructed well-established CEs to examine appropriate learning strategies, including how to generate training data among several shear tests and which optimization scheme to implement. We then tested whether an approximate CE could be obtained from a mesoscopic rheological model. Rheo-SINDy successfully identified correct CEs for the former case and derived approximate models that can reasonably predict nonlinear rheological properties for the latter case. Finally, we presented a method to obtain CEs satisfying rotational indifference, which is the basic principle of continuum mechanics. These results demonstrate that we can apply Rheo-SINDy to more complex rheological data.