Fitting tensorial constitutive models to experimental rheology using automatic differentiation

Accurately modeling the deformation and stress of fluid-like materials under complex forces and geometries is crucial for understanding and designing soft-matter systems. Experimental studies often report material responses as one-dimensional moduli, while theoretical studies develop tensorial evolution equations to describe the dynamics of stress and strain. Recently, neural networks have been used to represent unknown corrections to rheological equations or learn rheological models from data. These data-driven approaches create customized models that describe materials under different forces and geometries, even with limited training data, but they lack physical interpretability.

Building on the work of Lennon et. al. (2023), we present an alternative method to apply machine learning tools to fit experimental data to known physical models or more general tensorial basis models without using neural networks. The resulting representations offer increased physical interpretability and parameter efficiency, while still providing customizable constitutive laws for simulations in more complex geometries. This method allows us to evaluate the accuracy of different physical models applied to the same dataset, enabling tradeoffs between model complexity and improved accuracy.