Rheology-Informed Neural Networks (RhINNs) for forward and inverse metamodelling of complex fluids

Abstract

Reliable and accurate prediction of complex fluids’ response under flow is of great interest across many disciplines, from biological systems to virtually all soft materials. The challenge is to solve non-trivial time and rate dependent constitutive equations to describe these structured fluids under various flow protocols. We present Rheology-Informed Neural Networks (RhINNs) for solving systems of Ordinary Differential Equations (ODEs) adopted for complex fluids. The proposed RhINNs are employed to solve the constitutive models with multiple ODEs by benefiting from Automatic Differentiation in neural networks. In a direct solution, the RhINNs platform accurately predicts the fully resolved solution of constitutive equations for a Thixotropic-Elasto-Visco-Plastic (TEVP) complex fluid for a series of flow protocols. From a practical perspective, an exhaustive list of experiments are required to identify model parameters for a multi-variant constitutive TEVP model. RhINNs are found to learn these non-trivial model parameters for a complex material using a single flow protocol, enabling accurate modeling with limited number of experiments and at an unprecedented rate. We also show the RhINNs are not limited to a specific model and can be extended to include various models and recover complex manifestations of kinematic heterogeneities and transient shear banding of thixotropic fluids.

A rheologist's guideline to data-driven recovery of complex fluids' parameters from constitutive models

“…To explore the interplay between the neural net (referred to as NN in this work), the constitutive model (CM), and the data, it is best to start with the CM whose parameters are to be recovered. As a rheologically motivated example, a thixotropic elasto-visco-plastic (TEVP) constitutive model 8,32,42,43 is chosen, which consists of two coupled ODEs. In this formalism, eqn (1) describes the time evolution of the normalized shear stress, σ *( t ), which is the actual shear stress, σ ( t ), divided by the maximum shear stress of the sample ( σ max ):where the 〈·〉 superscript denotes the time derivative, G is the elastic modulus (in Pa), σ y is the yield stress (in Pa), η s and η p are the solvent (background) and plastic viscosities, respectively (in Pa s), ( t ), in s −1 , is the imposed shear rate (assuming a rate-controlled rheometry), and λ ( t ) is the dimensionless structure parameter .…”

“…This question was partially answered for fixed RhINN hyperparameters and a thixotropic elasto-visco-plastic (TEVP) model using only flow startup data during the training step. 32 Here, to answer this question more comprehensively, we relaxed the assumptions mentioned above. To this end, a set of flow startup data is generated, and the RhINN is requested to recover the fitting parameters of several simple-to-complex TEVP cases.…”

Digital rheometer twins: Learning the hidden rheology of complex fluids through rheology-informed graph neural networks

“…Learning biases can be softly penalized to favor of a specific solution or physical law ( 26 , 27 ), resulting in so-called physics-guided NNs. Alternatively, one can incorporate the physical laws of interest in the form of differential equations into the architecture of the NN, imposing a hard penalty ( 14 , 28 – 30 ) and construction of physics-informed NNs. While different in methodology, these pathways are not necessarily exclusive and can be used simultaneously.…”

“…Rheology-informed NNs can be devised by informing the ML platform of the underlying rheological constitutive models both implicitly ( 13 ) and explicitly ( 14 ). These NNs can also be integrated with conservation laws into computational fluid dynamics, and solved to provide velocity fields and fully resolved flow fields for complex fluids under different flows ( 15 ).…”

Rheological identification of jetted fluid using machine learning

“…More recently, physics-based ML algorithms, the so-called Physics-Informed Neural Network 23 (PINN), have been developed to reduce to diminish the need for big data sets by including governing physical laws in the ANN framework. This approach has then been extended to rheology by the Rheology-Informed Neural Networks (RhINNs) 24,25 which enables an accurate modeling of the rheological properties of a fluid with a limited number of experiments.…”