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Predicting the response of complex fluids to different flow conditions has been the focal point of rheology and is generally done via constitutive relations. There are, nonetheless, scenarios in which not much is known from the material mathematically, while data collection from samples is elusive, resource-intensive, or both. In such cases, meta-modeling of observables using a parametric surrogate model called multi-fidelity neural networks (MFNNs) may obviate the constitutive equation development step by leveraging only a handful of high-fidelity (Hi-Fi) data collected from experiments (or high-resolution simulations) and an abundance of low-fidelity (Lo-Fi) data generated synthetically to compensate for Hi-Fi data scarcity. To this end, MFNNs are employed to meta-model the material responses of a thermo-viscoelastic (TVE) fluid, consumer product Johnson’s® Baby Shampoo, under four flow protocols: steady shear, step growth, oscillatory, and small/large amplitude oscillatory shear (S/LAOS). In addition, the time–temperature superposition (TTS) of the material response and MFNN predictions are explored. By applying simple linear regression (without induction of any constitutive equation) on log-spaced Hi-Fi data, a series of Lo-Fi data were generated and found sufficient to obtain accurate material response recovery in terms of either interpolation or extrapolation for all flow protocols except for S/LAOS. This insufficiency is resolved by informing the MFNN platform with a linear constitutive model (Maxwell viscoelastic) resulting in simultaneous interpolation and extrapolation capabilities in S/LAOS material response recovery. The roles of data volume, flow type, and deformation range are discussed in detail, providing a practical pathway to multifidelity meta-modeling of different complex fluids.

Rheology-informed neural networks (RhINNs) have recently been popularized as data-driven platforms for solving rheologically relevant differential equations. While RhINNs can be employed to solve different constitutive equations of interest in a forward or inverse manner, their ability to do so strictly depends on the type of data and the choice of models embedded within their structure. Here, the applicability of RhINNs in general, and the interplay between the choice of models, parameters of the neural network itself, and the type of data at hand are studied. To do so, a RhINN is informed by a series of thixotropic elasto-visco-plastic (TEVP) constitutive models, and its ability to accurately recover model parameters from stress growth and oscillatory shear flow protocols is investigated. We observed that by simplifying the constitutive model, RhINN convergence is improved in terms of parameter recovery accuracy and computation speed while over-simplifying the model is detrimental to accuracy. Moreover, several hyperparameters, e.g., the learning rate, activation function, initial conditions for the fitting parameters, and error heuristics, should be at the top of the checklist when aiming to improve parameter recovery using RhINNs. Finally, the given data form plays a pivotal role, and no convergence is observed when one set of experiments is used as the given data for either of the flow protocols. The range of parameters is also a limiting factor when employing RhINNs for parameter recovery, and ad hoc modifications to the constitutive model can be trivial remedies to guarantee convergence when recovering fitting parameters with large values.

Developing constitutive models that can describe a complex fluid’s response to an applied stimulus has been one of the critical pursuits of rheologists. The complexity of the models typically goes hand-in-hand with that of the observed behaviors and can quickly become prohibitive depending on the choice of materials and/or flow protocols. Therefore, reducing the number of fitting parameters by seeking compact representations of those constitutive models can obviate extra experimentation to confine the parameter space. To this end, fractional derivatives in which the differential response of matter accepts non-integer orders have shown promise. Here, we develop neural networks that are informed by a series of different fractional constitutive models. These fractional rheology-informed neural networks (RhINNs) are then used to recover the relevant parameters (fractional derivative orders) of three fractional viscoelastic constitutive models, i.e., fractional Maxwell, Kelvin-Voigt, and Zener models. We find that for all three studied models, RhINNs recover the observed behavior accurately, although in some cases, the fractional derivative order is recovered with significant deviations from what is known as ground truth. This suggests that extra fractional elements are redundant when the material response is relatively simple. Therefore, choosing a fractional constitutive model for a given material response is contingent upon the response complexity, as fractional elements embody a wide range of transient material behaviors.