nn-PINNs: Non-Newtonian physics-informed neural networks for complex fluid modeling

Unbiased construction of constitutive relations for soft materials from experiments via rheology-informed neural networks

The ability to concisely describe the dynamical behavior of soft materials through closed-form constitutive relations holds the key to accelerated and informed design of materials and processes. The conventional approach is to construct constitutive relations through simplifying assumptions and approximating the time- and rate-dependent stress response of a complex fluid to an imposed deformation. While traditional frameworks have been foundational to our current understanding of soft materials, they often face a twofold existential limitation: i) Constructed on ideal and generalized assumptions, precise recovery of material-specific details is usually serendipitous, if possible, and ii) inherent biases that are involved by making those assumptions commonly come at the cost of new physical insight. This work introduces an approach by leveraging recent advances in scientific machine learning methodologies to discover the governing constitutive equation from experimental data for complex fluids. Our rheology-informed neural network framework is found capable of learning the hidden rheology of a complex fluid through a limited number of experiments. This is followed by construction of an unbiased material-specific constitutive relation that accurately describes a wide range of bulk dynamical behavior of the material. While extremely efficient in closed-form model discovery for a real-world complex system, the model also provides insight into the underpinning physics of the material.

Rotating Disk Electrodes beyond the Levich Approximation: Physics-Informed Neural Networks Reveal and Quantify Edge Effects

Physics-informed neural networks are used to characterize the mass transport to the rotating disk electrode (RDE), the most widely employed hydrodynamic electrode in electroanalysis. The PINN approach was first quantitatively verified via 1D simulations under the Levich approximation for cyclic voltammetry and chronoamperometry, allowing comparison of the results with finite difference simulations and analytical equations. However, the Levich approximation is only accurate for high Schmidt numbers (Sc > 1000). The PINN approach allowed consideration of smaller Sc, achieving an analytical level of accuracy (error <0.1%) comparable with independent numerical evaluation and confirming that the errors in the Levich equation can be as high as 3% when Sc = 1000 for rapidly diffusing species in aqueous solution. Entirely novel, the PINNs permit the solution of the 2D diffusion equation under cylindrical geometry incorporating radial diffusion and reveal the rotating disk electrode edge effect as a consequence of the nonuniform accessibility of the disc with greater currents flowing near the extremities. The contribution to the total current is quantified as a function of the rotation speed, disk radius, and analyte diffusion coefficient. The success in extending the theory for the rotating disk electrode beyond the Levich equation shows that PINNs can be an easier and more powerful substitute for conventional methods, both analytical and simulation based.

Scientific machine learning for modeling and simulating complex fluids

The formulation of rheological constitutive equations-models that relate internal stresses and deformations in complex fluids-is a critical step in the engineering of systems involving soft materials. While data-driven models provide accessible alternatives to expensive first-principles models and less accurate empirical models in many engineering disciplines, the development of similar models for complex fluids has lagged. The diversity of techniques for characterizing non-Newtonian fluid dynamics creates a challenge for classical machine learning approaches, which require uniformly structured training data. Consequently, early machine-learning based constitutive equations have not been portable between different deformation protocols or mechanical observables. Here, we present a data-driven framework that resolves such issues, allowing rheologists to construct learnable models that incorporate essential physical information, while remaining agnostic to details regarding particular experimental protocols or flow kinematics. These scientific machine learning models incorporate a universal approximator within a materially objective tensorial constitutive framework. By construction, these models respect physical constraints, such as frame-invariance and tensor symmetry, required by continuum mechanics. We demonstrate that this framework facilitates the rapid discovery of accurate constitutive equations from limited data and that the learned models may be used to describe more kinematically complex flows. This inherent flexibility admits the application of these "digital fluid twins" to a range of material systems and engineering problems. We illustrate this flexibility by deploying a trained model within a multidimensional computational fluid dynamics simulation-a task that is not achievable using any previously developed data-driven rheological equation of state.

A physics-informed neural network based on mixed data sampling for solving modified diffusion equations

We developed a physics-informed neural network based on a mixture of Cartesian grid sampling and Latin hypercube sampling to solve forward and backward modified diffusion equations. We optimized the parameters in the neural networks and the mixed data sampling by considering the squeeze boundary condition and the mixture coefficient, respectively. Then, we used a given modified diffusion equation as an example to demonstrate the efficiency of the neural network solver for forward and backward problems. The neural network results were compared with the numerical solutions, and good agreement with high accuracy was observed. This neural network solver can be generalized to other partial differential equations.

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

SignificanceScience-based data-driven methods that can describe the rheological behavior of complex fluids can be transformative across many disciplines. Digital rheometer twins, which are developed here, can significantly reduce the cost, time, and energy required to characterize complex fluids and predict their future behavior. This is made possible by combining two different methods of informing neural networks with the rheological underpinnings of a system, resulting in quantitative recovery of a gel's response to different flow protocols. The platform developed here is general enough that it can be extended to areas well beyond complex fluids modeling.