Finding the underlying viscoelastic constitutive equation via universal differential equations and differentiable physics

Abstract

Determining the appropriate constitutive model to describe the behavior of a given material is a fundamental, yet challenging, aspect of rheology. While data-driven methods present a promising path for refining these models, a more in-depth investigation into the capabilities and limitations of emerging techniques is required. This research addresses this gap by employing Universal Differential Equations (UDEs) and differentiable physics to model viscoelastic fluids, merging conventional differential equations with neural networks to reconstruct missing terms in constitutive models. This study focuses on analyzing four viscoelastic models, Upper Convected Maxwell (UCM), Johnson–Segalman, Giesekus, and Exponential Phan–Thien–Tanner (ePTT) using synthetic datasets. The methodology was tested across different experimental conditions, including oscillatory and startup flows. Relative error analyses revealed that the UDEs framework maintains low and stable errors (below 0.3%) for the UCM, Johnson–Segalman, and Giesekus models under various conditions, while exhibiting higher but consistent errors (4%) for the ePTT model due to its strong nonlinearity. These findings highlight the potential of UDEs in fluid mechanics while also identifying critical areas for methodological improvement. Additionally, a model distillation approach was employed to extract simplified models from complex ones, emphasizing the versatility and robustness of UDEs in rheological modeling.